wpmath0000002_11

LB.html

  1. l b ( n ) = l o g 2 ( n ) lb(n)=log_{2}(n)

LD.html

  1. ld ( x ) = log 2 ( x ) \mathrm{ld}(x)=\log_{2}(x)

Le_Sage's_theory_of_gravitation.html

  1. S ρ v S\sqrt{\rho}v

Least_squares.html

  1. ( x i , y i ) (x_{i},y_{i})\!
  2. x i x_{i}\!
  3. y i y_{i}\!
  4. f ( x , β ) f(x,\beta)
  5. s y m b o l β symbol\beta
  6. S = i = 1 n r i 2 S=\sum_{i=1}^{n}{r_{i}}^{2}
  7. r i = y i - f ( x i , s y m b o l β ) . r_{i}=y_{i}-f(x_{i},symbol\beta).
  8. β 0 \beta_{0}
  9. β 1 \beta_{1}
  10. f ( x , s y m b o l β ) = β 0 + β 1 x f(x,symbol\beta)=\beta_{0}+\beta_{1}x
  11. S β j = 2 i r i r i β j = 0 , j = 1 , , m , \frac{\partial S}{\partial\beta_{j}}=2\sum_{i}r_{i}\frac{\partial r_{i}}{% \partial\beta_{j}}=0,\ j=1,\ldots,m,
  12. r i = y i - f ( x i , s y m b o l β ) r_{i}=y_{i}-f(x_{i},symbol\beta)
  13. - 2 i r i f ( x i , s y m b o l β ) β j = 0 , j = 1 , , m . -2\sum_{i}r_{i}\frac{\partial f(x_{i},symbol\beta)}{\partial\beta_{j}}=0,\ j=1% ,\ldots,m.
  14. f ( x , β ) = j = 1 m β j ϕ j ( x ) , f(x,\beta)=\sum_{j=1}^{m}\beta_{j}\phi_{j}(x),
  15. ϕ j \phi_{j}
  16. x x
  17. X i j = f ( x i , s y m b o l β ) β j = ϕ j ( x i ) , X_{ij}=\frac{\partial f(x_{i},symbol\beta)}{\partial\beta_{j}}=\phi_{j}(x_{i}),
  18. s y m b o l β symbol\beta
  19. s y m b o l β ^ = ( X T X ) - 1 X T s y m b o l y . symbol{\hat{\beta}}=(X^{T}X)^{-1}X^{T}symboly.
  20. β \beta
  21. β j k + 1 = β j k + Δ β j , {\beta_{j}}^{k+1}={\beta_{j}}^{k}+\Delta\beta_{j},
  22. Δ β j \Delta\beta_{j}
  23. s y m b o l β k symbol\beta^{k}
  24. f ( x i , s y m b o l β ) \displaystyle f(x_{i},symbol\beta)
  25. r i = y i - f k ( x i , s y m b o l β ) - k = 1 m J i k Δ β k = Δ y i - j = 1 m J i j Δ β j . r_{i}=y_{i}-f^{k}(x_{i},symbol\beta)-\sum_{k=1}^{m}J_{ik}\Delta\beta_{k}=% \Delta y_{i}-\sum_{j=1}^{m}J_{ij}\Delta\beta_{j}.
  26. r i r_{i}
  27. Δ β j \Delta\beta_{j}
  28. - 2 i = 1 n J i j ( Δ y i - k = 1 m J i k Δ β k ) = 0 , -2\sum_{i=1}^{n}J_{ij}\left(\Delta y_{i}-\sum_{k=1}^{m}J_{ik}\Delta\beta_{k}% \right)=0,
  29. i = 1 n k = 1 m J i j J i k Δ β k = i = 1 n J i j Δ y i ( j = 1 , , m ) . \sum_{i=1}^{n}\sum_{k=1}^{m}J_{ij}J_{ik}\Delta\beta_{k}=\sum_{i=1}^{n}J_{ij}% \Delta y_{i}\qquad(j=1,\ldots,m).
  30. ( 𝐉 𝐓 𝐉 ) 𝚫 𝐬𝐲𝐦𝐛𝐨𝐥 β = 𝐉 𝐓 𝚫 𝐲 . \mathbf{\left(J^{T}J\right)\Delta symbol\beta=J^{T}\Delta y}.\,
  31. f = X i 1 β 1 + X i 2 β 2 + f=X_{i1}\beta_{1}+X_{i2}\beta_{2}+\cdots
  32. β 2 , e β x \beta^{2},e^{\beta x}
  33. f / β j \partial f/\partial\beta_{j}
  34. y y
  35. y = f ( F , k ) = k F y=f(F,k)=kF\!
  36. ( F i , y i ) , i = 1 , , n (F_{i},y_{i}),\ i=1,\dots,n\!
  37. ε \varepsilon
  38. y i = k F i + ε i . y_{i}=kF_{i}+\varepsilon_{i}.\,
  39. S = i = 1 n ( y i - k F i ) 2 . S=\sum_{i=1}^{n}\left(y_{i}-kF_{i}\right)^{2}.
  40. k ^ = i F i y i i F i 2 . \hat{k}=\frac{\sum_{i}F_{i}y_{i}}{\sum_{i}{F_{i}}^{2}}.
  41. var ( β ^ j ) \operatorname{var}(\hat{\beta}_{j})
  42. var ( β ^ j ) = σ 2 ( [ X T X ] - 1 ) j j S n - m ( [ X T X ] - 1 ) j j , \,\text{var}(\hat{\beta}_{j})=\sigma^{2}\left(\left[X^{T}X\right]^{-1}\right)_% {jj}\approx\frac{S}{n-m}\left(\left[X^{T}X\right]^{-1}\right)_{jj},
  43. s y m b o l β ^ \hat{symbol{\beta}}
  44. s y m b o l β ^ \hat{symbol{\beta}}
  45. S = i = 1 n W i i r i 2 , W i i = 1 σ i 2 S=\sum_{i=1}^{n}W_{ii}{r_{i}}^{2},\qquad W_{ii}=\frac{1}{{\sigma_{i}}^{2}}
  46. - 2 i W i i f ( x i , s y m b o l β ) β j r i = 0 , j = 1 , , n -2\sum_{i}W_{ii}\frac{\partial f(x_{i},symbol{\beta})}{\partial\beta_{j}}r_{i}% =0,\qquad j=1,\ldots,n
  47. i = 1 n k = 1 m X i j W i i X i k β ^ k = i = 1 n X i j W i i y i , j = 1 , , m . \sum_{i=1}^{n}\sum_{k=1}^{m}X_{ij}W_{ii}X_{ik}\hat{\beta}_{k}=\sum_{i=1}^{n}X_% {ij}W_{ii}y_{i},\qquad j=1,\ldots,m\,.
  48. ( 𝐗 𝐓 𝐖𝐗 ) 𝐬𝐲𝐦𝐛𝐨𝐥 β ^ = 𝐗 𝐓 𝐖𝐲 . \mathbf{\left(X^{T}WX\right)\hat{symbol{\beta}}=X^{T}Wy}.
  49. 𝐰 𝐢𝐢 = 𝐖 𝐢𝐢 \mathbf{w_{ii}}=\sqrt{\mathbf{W_{ii}}}
  50. ( 𝐗 𝐓 𝐗 ) 𝐬𝐲𝐦𝐛𝐨𝐥 β ^ = 𝐗 𝐓 𝐲 \mathbf{\left(X^{\prime T}X^{\prime}\right)\hat{symbol{\beta}}=X^{\prime T}y^{% \prime}}\,
  51. 𝐗 = 𝐰𝐗 , 𝐲 = 𝐰𝐲 . \mathbf{X^{\prime}}=\mathbf{wX},\mathbf{y^{\prime}}=\mathbf{wy}.\,
  52. ( 𝐉 𝐓 𝐖𝐉 ) 𝐬𝐲𝐦𝐛𝐨𝐥 𝚫 β = 𝐉 𝐓 𝐖𝐬𝐲𝐦𝐛𝐨𝐥 𝚫 𝐲 . \mathbf{\left(J^{T}WJ\right)symbol\Delta\beta=J^{T}Wsymbol\Delta y}.\,
  53. y y
  54. β 2 \|\beta\|^{2}
  55. α β 2 \alpha\|\beta\|^{2}
  56. α \alpha
  57. β 1 \|\beta\|_{1}
  58. α β 1 \alpha\|\beta\|_{1}
  59. α \alpha

Legendre_function.html

  1. ( 1 - x 2 ) y ′′ - 2 x y + [ λ ( λ + 1 ) - μ 2 1 - x 2 ] y = 0 , (1-x^{2})\,y^{\prime\prime}-2xy^{\prime}+\left[\lambda(\lambda+1)-\frac{\mu^{2% }}{1-x^{2}}\right]\,y=0,\,
  2. P λ μ ( z ) = 1 Γ ( 1 - μ ) [ 1 + z 1 - z ] 2 μ / 2 F 1 ( - λ , λ + 1 ; 1 - μ ; 1 - z 2 ) , for | 1 - z | < 2 P_{\lambda}^{\mu}(z)=\frac{1}{\Gamma(1-\mu)}\left[\frac{1+z}{1-z}\right]^{\mu/% 2}\,_{2}F_{1}(-\lambda,\lambda+1;1-\mu;\frac{1-z}{2}),\qquad\,\text{for }\ |1-% z|<2
  3. Γ \Gamma
  4. F 1 2 {}_{2}F_{1}
  5. Q λ μ ( z ) Q_{\lambda}^{\mu}(z)
  6. Q λ μ ( z ) = π Γ ( λ + μ + 1 ) 2 λ + 1 Γ ( λ + 3 / 2 ) e i μ π ( z 2 - 1 ) μ / 2 z λ + μ + 1 2 F 1 ( λ + μ + 1 2 , λ + μ + 2 2 ; λ + 3 2 ; 1 z 2 ) , for | z | > 1. Q_{\lambda}^{\mu}(z)=\frac{\sqrt{\pi}\ \Gamma(\lambda+\mu+1)}{2^{\lambda+1}% \Gamma(\lambda+3/2)}\frac{e^{i\mu\pi}(z^{2}-1)^{\mu/2}}{z^{\lambda+\mu+1}}\,_{% 2}F_{1}\left(\frac{\lambda+\mu+1}{2},\frac{\lambda+\mu+2}{2};\lambda+\frac{3}{% 2};\frac{1}{z^{2}}\right),\qquad\,\text{for}\ \ |z|>1.
  7. P λ ( x ) P_{\lambda}(x)
  8. P λ μ ( x ) P^{\mu}_{\lambda}(x)
  9. P λ ( z ) = 1 2 π i 1 , z ( t 2 - 1 ) λ 2 λ ( t - z ) λ + 1 d t P_{\lambda}(z)=\frac{1}{2\pi i}\int_{1,z}\frac{(t^{2}-1)^{\lambda}}{2^{\lambda% }(t-z)^{\lambda+1}}dt
  10. P s ( x ) = 1 2 π - π π ( x + x 2 - 1 cos θ ) s d θ = 1 π 0 1 ( x + x 2 - 1 ( 2 t - 1 ) ) s d t t ( 1 - t ) , s P_{s}(x)=\frac{1}{2\pi}\int_{-\pi}^{\pi}\left(x+\sqrt{x^{2}-1}\cos\theta\right% )^{s}d\theta=\frac{1}{\pi}\int_{0}^{1}\left(x+\sqrt{x^{2}-1}(2t-1)\right)^{s}% \frac{dt}{\sqrt{t(1-t)}},\qquad s\in\mathbb{C}
  11. P s P_{s}
  12. L 1 ( G / / K ) L^{1}(G//K)
  13. G / / K G//K
  14. S L ( 2 , ) SL(2,\mathbb{R})
  15. L 1 ( G / / K ) L^{1}(G//K)
  16. L 1 ( G / / K ) f f ^ L^{1}(G//K)\ni f\mapsto\hat{f}
  17. f ^ ( s ) = 1 f ( x ) P s ( x ) d x , - 1 ( s ) 0 \hat{f}(s)=\int_{1}^{\infty}f(x)P_{s}(x)dx,\qquad-1\leq\Re(s)\leq 0

Legendre_polynomials.html

  1. d d x [ ( 1 - x 2 ) d d x P n ( x ) ] + n ( n + 1 ) P n ( x ) = 0. {d\over dx}\left[(1-x^{2}){d\over dx}P_{n}(x)\right]+n(n+1)P_{n}(x)=0.
  2. P n ( x ) = 1 2 n n ! d n d x n [ ( x 2 - 1 ) n ] . P_{n}(x)={1\over 2^{n}n!}{d^{n}\over dx^{n}}\left[(x^{2}-1)^{n}\right].
  3. ( x 2 - 1 ) d d x ( x 2 - 1 ) n = 2 n x ( x 2 - 1 ) n (x^{2}-1)\frac{d}{dx}(x^{2}-1)^{n}=2nx(x^{2}-1)^{n}
  4. 1 1 - 2 x t + t 2 = n = 0 P n ( x ) t n . \frac{1}{\sqrt{1-2xt+t^{2}}}=\sum_{n=0}^{\infty}P_{n}(x)t^{n}.
  5. P 0 ( x ) = 1 , P 1 ( x ) = x P_{0}(x)=1,\quad P_{1}(x)=x
  6. x - t 1 - 2 x t + t 2 = ( 1 - 2 x t + t 2 ) n = 1 n P n ( x ) t n - 1 . \frac{x-t}{\sqrt{1-2xt+t^{2}}}=(1-2xt+t^{2})\sum_{n=1}^{\infty}nP_{n}(x)t^{n-1}.
  7. ( n + 1 ) P n + 1 ( x ) = ( 2 n + 1 ) x P n ( x ) - n P n - 1 ( x ) . (n+1)P_{n+1}(x)=(2n+1)xP_{n}(x)-nP_{n-1}(x).\,
  8. P n ( x ) = 1 2 n k = 0 n ( n k ) 2 ( x - 1 ) n - k ( x + 1 ) k = k = 0 n ( n k ) ( - n - 1 k ) ( 1 - x 2 ) k = 2 n k = 0 n x k ( n k ) ( n + k - 1 2 n ) , \begin{aligned}\displaystyle P_{n}(x)&\displaystyle=\frac{1}{2^{n}}\sum_{k=0}^% {n}{n\choose k}^{2}(x-1)^{n-k}(x+1)^{k}\\ &\displaystyle=\sum_{k=0}^{n}{n\choose k}{-n-1\choose k}\left(\frac{1-x}{2}% \right)^{k}\\ &\displaystyle=2^{n}\cdot\sum_{k=0}^{n}x^{k}{n\choose k}{\frac{n+k-1}{2}% \choose n},\end{aligned}
  9. n P n ( x ) 0 1 1 x 2 1 2 ( 3 x 2 - 1 ) 3 1 2 ( 5 x 3 - 3 x ) 4 1 8 ( 35 x 4 - 30 x 2 + 3 ) 5 1 8 ( 63 x 5 - 70 x 3 + 15 x ) 6 1 16 ( 231 x 6 - 315 x 4 + 105 x 2 - 5 ) 7 1 16 ( 429 x 7 - 693 x 5 + 315 x 3 - 35 x ) 8 1 128 ( 6435 x 8 - 12012 x 6 + 6930 x 4 - 1260 x 2 + 35 ) 9 1 128 ( 12155 x 9 - 25740 x 7 + 18018 x 5 - 4620 x 3 + 315 x ) 10 1 256 ( 46189 x 10 - 109395 x 8 + 90090 x 6 - 30030 x 4 + 3465 x 2 - 63 ) \begin{array}[]{ccl}n&&P_{n}(x)\\ \hline 0&&1\\ 1&&x\\ 2&&\frac{1}{2}(3x^{2}-1)\\ 3&&\frac{1}{2}(5x^{3}-3x)\\ 4&&\frac{1}{8}(35x^{4}-30x^{2}+3)\\ 5&&\frac{1}{8}(63x^{5}-70x^{3}+15x)\\ 6&&\frac{1}{16}(231x^{6}-315x^{4}+105x^{2}-5)\\ 7&&\frac{1}{16}(429x^{7}-693x^{5}+315x^{3}-35x)\\ 8&&\frac{1}{128}(6435x^{8}-12012x^{6}+6930x^{4}-1260x^{2}+35)\\ 9&&\frac{1}{128}(12155x^{9}-25740x^{7}+18018x^{5}-4620x^{3}+315x)\\ 10&&\frac{1}{256}(46189x^{10}-109395x^{8}+90090x^{6}-30030x^{4}+3465x^{2}-63)% \\ \hline\end{array}
  10. - 1 1 P m ( x ) P n ( x ) d x = 2 2 n + 1 δ m n \int_{-1}^{1}P_{m}(x)P_{n}(x)\,dx={2\over{2n+1}}\delta_{mn}
  11. d d x [ ( 1 - x 2 ) d d x P ( x ) ] = - λ P ( x ) , {d\over dx}\left[(1-x^{2}){d\over dx}P(x)\right]=-\lambda P(x),
  12. 1 | 𝐱 - 𝐱 | = 1 r 2 + r 2 - 2 r r cos γ = = 0 r r + 1 P ( cos γ ) \frac{1}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}=\frac{1}{\sqrt{r^{2}+r^{% \prime 2}-2rr^{\prime}\cos\gamma}}=\sum_{\ell=0}^{\infty}\frac{r^{\prime\ell}}% {r^{\ell+1}}P_{\ell}(\cos\gamma)
  13. r r
  14. r r^{\prime}
  15. 𝐱 \mathbf{x}
  16. 𝐱 \mathbf{x}^{\prime}
  17. γ \gamma
  18. r > r r>r^{\prime}
  19. 2 Φ ( 𝐱 ) = 0 \nabla^{2}\Phi(\mathbf{x})=0
  20. 𝐳 ^ \widehat{\mathbf{z}}
  21. θ \theta
  22. 𝐳 ^ \widehat{\mathbf{z}}
  23. Φ ( r , θ ) = = 0 [ A r + B r - ( + 1 ) ] P ( cos θ ) . \Phi(r,\theta)=\sum_{\ell=0}^{\infty}\left[A_{\ell}r^{\ell}+B_{\ell}r^{-(\ell+% 1)}\right]P_{\ell}(\cos\theta).
  24. A A_{\ell}
  25. B B_{\ell}
  26. 1 1 + η 2 - 2 η x = k = 0 η k P k ( x ) \frac{1}{\sqrt{1+\eta^{2}-2\eta x}}=\sum_{k=0}^{\infty}\eta^{k}P_{k}(x)
  27. Φ ( r , θ ) \Phi(r,\theta)
  28. z = a z=a
  29. Φ ( r , θ ) 1 R = 1 r 2 + a 2 - 2 a r cos θ . \Phi(r,\theta)\propto\frac{1}{R}=\frac{1}{\sqrt{r^{2}+a^{2}-2ar\cos\theta}}.
  30. Φ ( r , θ ) 1 r k = 0 ( a r ) k P k ( cos θ ) \Phi(r,\theta)\propto\frac{1}{r}\sum_{k=0}^{\infty}\left(\frac{a}{r}\right)^{k% }P_{k}(\cos\theta)
  31. T n ( cos θ ) cos n θ T_{n}(\cos\theta)\equiv\cos n\theta
  32. P n ( cos θ ) P_{n}(\cos\theta)
  33. T 0 ( cos θ ) = 1 = P 0 ( cos θ ) T_{0}(\cos\theta)=1=P_{0}(\cos\theta)
  34. T 1 ( cos θ ) = cos θ = P 1 ( cos θ ) T_{1}(\cos\theta)=\cos\theta=P_{1}(\cos\theta)
  35. T 2 ( cos θ ) = cos 2 θ = 1 3 ( 4 P 2 ( cos θ ) - P 0 ( cos θ ) ) T_{2}(\cos\theta)=\cos 2\theta=\frac{1}{3}(4P_{2}(\cos\theta)-P_{0}(\cos\theta))
  36. T 3 ( cos θ ) = cos 3 θ = 1 5 ( 8 P 3 ( cos θ ) - 3 P 1 ( cos θ ) ) T_{3}(\cos\theta)=\cos 3\theta=\frac{1}{5}(8P_{3}(\cos\theta)-3P_{1}(\cos% \theta))
  37. T 4 ( cos θ ) = cos 4 θ = 1 105 ( 192 P 4 ( cos θ ) - 80 P 2 ( cos θ ) - 7 P 0 ( cos θ ) ) T_{4}(\cos\theta)=\cos 4\theta=\frac{1}{105}(192P_{4}(\cos\theta)-80P_{2}(\cos% \theta)-7P_{0}(\cos\theta))
  38. T 5 ( cos θ ) = cos 5 θ = 1 63 ( 128 P 5 ( cos θ ) - 56 P 3 ( cos θ ) - 9 P 1 ( cos θ ) ) T_{5}(\cos\theta)=\cos 5\theta=\frac{1}{63}(128P_{5}(\cos\theta)-56P_{3}(\cos% \theta)-9P_{1}(\cos\theta))
  39. T 6 ( cos θ ) = cos 6 θ = 1 1155 ( 2560 P 6 ( cos θ ) - 1152 P 4 ( cos θ ) - 220 P 2 ( cos θ ) - 33 P 0 ( cos θ ) ) T_{6}(\cos\theta)=\cos 6\theta=\frac{1}{1155}(2560P_{6}(\cos\theta)-1152P_{4}(% \cos\theta)-220P_{2}(\cos\theta)-33P_{0}(\cos\theta))
  40. sin ( n + 1 ) θ \sin(n+1)\theta
  41. sin ( n + 1 ) θ sin θ = = 0 n P ( cos θ ) P n - ( cos θ ) \frac{\sin(n+1)\theta}{\sin\theta}=\sum_{\ell=0}^{n}P_{\ell}(\cos\theta)P_{n-% \ell}(\cos\theta)
  42. P n ( - x ) = ( - 1 ) n P n ( x ) . P_{n}(-x)=(-1)^{n}P_{n}(x).\,
  43. P n ( 1 ) = 1. P_{n}(1)=1.\,
  44. P n ( 1 ) = n ( n + 1 ) 2 . P_{n}^{\prime}(1)=\frac{n(n+1)}{2}.\,
  45. ( n + 1 ) P n + 1 ( x ) = ( 2 n + 1 ) x P n ( x ) - n P n - 1 ( x ) (n+1)P_{n+1}(x)=(2n+1)xP_{n}(x)-nP_{n-1}(x)\,
  46. x 2 - 1 n d d x P n ( x ) = x P n ( x ) - P n - 1 ( x ) . {x^{2}-1\over n}{d\over dx}P_{n}(x)=xP_{n}(x)-P_{n-1}(x).
  47. ( 2 n + 1 ) P n ( x ) = d d x [ P n + 1 ( x ) - P n - 1 ( x ) ] . (2n+1)P_{n}(x)={d\over dx}\left[P_{n+1}(x)-P_{n-1}(x)\right].
  48. d d x P n + 1 ( x ) = ( 2 n + 1 ) P n ( x ) + ( 2 ( n - 2 ) + 1 ) P n - 2 ( x ) + ( 2 ( n - 4 ) + 1 ) P n - 4 ( x ) + {d\over dx}P_{n+1}(x)=(2n+1)P_{n}(x)+(2(n-2)+1)P_{n-2}(x)+(2(n-4)+1)P_{n-4}(x)+\ldots
  49. d d x P n + 1 ( x ) = 2 P n ( x ) P n ( x ) 2 + 2 P n - 2 ( x ) P n - 2 ( x ) 2 + {d\over dx}P_{n+1}(x)={2P_{n}(x)\over\|P_{n}(x)\|^{2}}+{2P_{n-2}(x)\over\|P_{n% -2}(x)\|^{2}}+\ldots
  50. P n ( x ) \|P_{n}(x)\|
  51. P n ( x ) = - 1 1 ( P n ( x ) ) 2 d x = 2 2 n + 1 . \|P_{n}(x)\|=\sqrt{\int_{-1}^{1}(P_{n}(x))^{2}\,dx}=\sqrt{\frac{2}{2n+1}}.
  52. P n ( x ) = k = 0 n ( - 1 ) k ( n k ) 2 ( 1 + x 2 ) n - k ( 1 - x 2 ) k . P_{n}(x)=\sum_{k=0}^{n}(-1)^{k}\begin{pmatrix}n\\ k\end{pmatrix}^{2}\left(\frac{1+x}{2}\right)^{n-k}\left(\frac{1-x}{2}\right)^{% k}.
  53. j = 0 n P j ( x ) 0 ( x - 1 ) . \sum_{j=0}^{n}P_{j}(x)\geq 0\qquad(x\geq-1).
  54. - 1 y 1 -1\leq y\leq 1
  55. - 1 x 1 -1\leq x\leq 1
  56. δ ( y - x ) = 1 2 = 0 ( 2 + 1 ) P ( y ) P ( x ) . \delta(y-x)=\frac{1}{2}\sum_{\ell=0}^{\infty}(2\ell+1)P_{\ell}(y)P_{\ell}(x)\,.
  57. P ( r r ) = 4 π 2 + 1 m = - Y m ( θ , ϕ ) Y m * ( θ , ϕ ) . P_{\ell}({r}\cdot{r^{\prime}})=\frac{4\pi}{2\ell+1}\sum_{m=-\ell}^{\ell}Y_{% \ell m}(\theta,\phi)Y_{\ell m}^{*}(\theta^{\prime},\phi^{\prime})\,.
  58. ( θ , ϕ ) (\theta,\phi)
  59. ( θ , ϕ ) (\theta^{\prime},\phi^{\prime})
  60. \ell\rightarrow\infty
  61. P ( cos θ ) = J 0 ( θ ) + 𝒪 ( - 1 ) = 2 2 π sin θ cos [ ( + 1 2 ) θ - π 4 ] + 𝒪 ( - 1 ) P_{\ell}(\cos\theta)=J_{0}(\ell\theta)+\mathcal{O}(\ell^{-1})=\frac{2}{\sqrt{2% \pi\ell\sin\theta}}\cos\left[\left(\ell+\frac{1}{2}\right)\theta-\frac{\pi}{4}% \right]+\mathcal{O}(\ell^{-1})
  62. P ( 1 1 - e 2 ) = I 0 ( e ) + 𝒪 ( - 1 ) = 1 2 π e ( 1 + e ) ( + 1 ) / 2 ( 1 - e ) / 2 + 𝒪 ( - 1 ) , P_{\ell}\left(\frac{1}{\sqrt{1-e^{2}}}\right)=I_{0}(\ell e)+\mathcal{O}(\ell^{% -1})=\frac{1}{\sqrt{2\pi\ell e}}\frac{(1+e)^{(\ell+1)/2}}{(1-e)^{\ell/2}}+% \mathcal{O}(\ell^{-1})\,,
  63. J 0 J_{0}
  64. I 0 I_{0}
  65. P n ~ ( x ) = P n ( 2 x - 1 ) \tilde{P_{n}}(x)=P_{n}(2x-1)
  66. x 2 x - 1 x\mapsto 2x-1
  67. P n ~ ( x ) \tilde{P_{n}}(x)
  68. 0 1 P m ~ ( x ) P n ~ ( x ) d x = 1 2 n + 1 δ m n . \int_{0}^{1}\tilde{P_{m}}(x)\tilde{P_{n}}(x)\,dx={1\over{2n+1}}\delta_{mn}.
  69. P n ~ ( x ) = ( - 1 ) n k = 0 n ( n k ) ( n + k k ) ( - x ) k . \tilde{P_{n}}(x)=(-1)^{n}\sum_{k=0}^{n}{n\choose k}{n+k\choose k}(-x)^{k}.
  70. P n ~ ( x ) = 1 n ! d n d x n [ ( x 2 - x ) n ] . \tilde{P_{n}}(x)=\frac{1}{n!}{d^{n}\over dx^{n}}\left[(x^{2}-x)^{n}\right].\,
  71. P n ~ ( x ) \tilde{P_{n}}(x)
  72. 2 x - 1 2x-1
  73. 6 x 2 - 6 x + 1 6x^{2}-6x+1
  74. 20 x 3 - 30 x 2 + 12 x - 1 20x^{3}-30x^{2}+12x-1
  75. 70 x 4 - 140 x 3 + 90 x 2 - 20 x + 1 70x^{4}-140x^{3}+90x^{2}-20x+1
  76. ( Q n ) (Q_{n})
  77. Q n ( x ) Q_{n}(x)
  78. Q n ( x ) = n ! 1 3 ( 2 n + 1 ) [ x - ( n + 1 ) + ( n + 1 ) ( n + 2 ) 2 ( n + 3 ) x - ( n + 3 ) + ( n + 1 ) ( n + 2 ) ( n + 3 ) ( n + 4 ) 2 4 ( 2 n + 3 ) ( 2 n + 5 ) x - ( n + 5 ) + ] Q_{n}(x)=\frac{n!}{1\cdot 3\cdots(2n+1)}\left[x^{-(n+1)}+\frac{(n+1)(n+2)}{2(n% +3)}x^{-(n+3)}+\frac{(n+1)(n+2)(n+3)(n+4)}{2\cdot 4(2n+3)(2n+5)}x^{-(n+5)}+% \cdots\right]
  79. d d x [ ( 1 - x 2 ) d d x f ( x ) ] + n ( n + 1 ) f ( x ) = 0 {d\over dx}\left[(1-x^{2}){d\over dx}f(x)\right]+n(n+1)f(x)=0
  80. f ( x ) = A P n ( x ) + B Q n ( x ) f(x)=AP_{n}(x)+BQ_{n}(x)

Legume.html

  1. N 2 + 8 H + + 8 e - 2 N H 3 + H 2 N_{2}+8H^{+}+8e^{-}\to 2NH_{3}+H_{2}
  2. N H 3 + H + N H 4 + NH_{3}+H^{+}\to NH_{4}^{+}

Leidenfrost_effect.html

  1. q A m i n = C h f g ρ v [ σ g ( ρ L - ρ v ) ( ρ L + ρ v ) 2 ] 1 4 {{\frac{q}{A}}_{min}}=C{{h}_{fg}}{{\rho}_{v}}{{\left[\frac{\sigma g\left({{% \rho}_{L}}-{{\rho}_{v}}\right)}{{{\left({{\rho}_{L}}+{{\rho}_{v}}\right)}^{2}}% }\right]}^{{}^{1}\!\!\diagup\!\!{}_{4}\;}}
  2. h = C [ k v 3 ρ v g ( ρ L - ρ v ) ( h f g + 0.4 c p v ( T s - T s a t ) ) D o μ v ( T s - T s a t ) ] 1 4 h=C{{\left[\frac{k_{v}^{3}{{\rho}_{v}}g\left({{\rho}_{L}}-{{\rho}_{v}}\right)% \left({{h}_{fg}}+0.4{{c}_{pv}}\left({{T}_{s}}-{{T}_{sat}}\right)\right)}{{{D}_% {o}}{{\mu}_{v}}\left({{T}_{s}}-{{T}_{sat}}\right)}\right]}^{{}^{1}\!\!\diagup% \!\!{}_{4}\;}}
  3. D o {{D}_{o}}
  4. h = 0.325 [ k v f 3 ρ v f g ( ρ L - ρ v ) ( h f g + 0.4 c p v ( T s - T s a t ) ) μ v f ( T s - T s a t ) σ / g ( ρ L - ρ v ) ] 1 4 h=0.325{{\left[\frac{k_{vf}^{3}{{\rho}_{vf}}g\left({{\rho}_{L}}-{{\rho}_{v}}% \right)\left({{h}_{fg}}+0.4{{c}_{pv}}\left({{T}_{s}}-{{T}_{sat}}\right)\right)% }{{{\mu}_{vf}}\left({{T}_{s}}-{{T}_{sat}}\right)\sqrt{\sigma/g\left({{\rho}_{L% }}-{{\rho}_{v}}\right)}}\right]}^{{}^{1}\!\!\diagup\!\!{}_{4}\;}}
  5. h [ μ v 2 g ρ v ( ρ L - ρ v ) k v 3 ] 1 3 = 0.0020 [ 4 m π D v μ v ] 0.6 h{{\left[\frac{\mu_{v}^{2}}{g{{\rho}_{v}}\left({{\rho}_{L}}-{{\rho}_{v}}\right% )k_{v}^{3}}\right]}^{{}^{1}\!\!\diagup\!\!{}_{3}\;}}=0.0020{{\left[\frac{4m}{% \pi{{D}_{v}}{{\mu}_{v}}}\right]}^{0.6}}
  6. l b m / h r l{{b}_{m}}/hr
  7. h 4 3 = h c o n v 4 3 + h r a d h 1 3 {{h}^{{}^{4}\!\!\diagup\!\!{}_{3}\;}}={{h}_{conv}}^{{}^{4}\!\!\diagup\!\!{}_{3% }\;}+{{h}_{rad}}{{h}^{{}^{1}\!\!\diagup\!\!{}_{3}\;}}
  8. h r a d < h c o n v {{h}_{rad}}<{{h}_{conv}}
  9. h = h c o n v + 3 4 h r a d h={{h}_{conv}}+\frac{3}{4}{{h}_{rad}}
  10. h r a d {{h}_{rad}}
  11. h r a d = ε σ ( T s 4 - T s a t 4 ) ( T s - T s a t ) {{h}_{rad}}=\frac{\varepsilon\sigma\left(T_{s}^{4}-T_{sat}^{4}\right)}{\left({% {T}_{s}}-{{T}_{sat}}\right)}
  12. ε \varepsilon
  13. σ \sigma

Lemniscate_of_Bernoulli.html

  1. ( x 2 + y 2 ) 2 = 2 a 2 ( x 2 - y 2 ) (x^{2}+y^{2})^{2}=2a^{2}(x^{2}-y^{2})\,
  2. r 2 = 2 a 2 cos 2 θ r^{2}=2a^{2}\cos 2\theta\,
  3. x = a 2 cos ( t ) sin 2 ( t ) + 1 ; y = a 2 cos ( t ) sin ( t ) sin 2 ( t ) + 1 x=\frac{a\sqrt{2}\cos(t)}{\sin^{2}(t)+1};\qquad y=\frac{a\sqrt{2}\cos(t)\sin(t% )}{\sin^{2}(t)+1}
  4. r r = a 2 rr^{\prime}=a^{2}\,
  5. Q = 2 s - 1 Q=2s-1\,

Lennard-Jones_potential.html

  1. V L J = 4 ε [ ( σ r ) 12 - ( σ r ) 6 ] = ε [ ( r m r ) 12 - 2 ( r m r ) 6 ] , V_{LJ}=4\varepsilon\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma% }{r}\right)^{6}\right]=\varepsilon\left[\left(\frac{r_{m}}{r}\right)^{12}-2% \left(\frac{r_{m}}{r}\right)^{6}\right],
  2. V B = γ [ e - r / r 0 - ( r 0 r ) 6 ] . V_{B}=\gamma\left[e^{-r/r_{0}}-\left(\frac{r_{0}}{r}\right)^{6}\right].
  3. V L J ( r ) = A r 12 - B r 6 , V_{LJ}(r)=\frac{A}{r^{12}}-\frac{B}{r^{6}},
  4. A = 4 ε σ 12 A=4\varepsilon\sigma^{12}
  5. B = 4 ε σ 6 B=4\varepsilon\sigma^{6}
  6. σ = A B 6 \sigma=\sqrt[6]{\frac{A}{B}}
  7. ε = B 2 4 A \varepsilon=\frac{B^{2}}{4A}
  8. V L J ( r c ) = V L J ( 2.5 σ ) = 4 ε [ ( σ 2.5 σ ) 12 - ( σ 2.5 σ ) 6 ] - 0.0163 ε \displaystyle V_{LJ}(r_{c})=V_{LJ}(2.5\sigma)=4\varepsilon\left[\left(\frac{% \sigma}{2.5\sigma}\right)^{12}-\left(\frac{\sigma}{2.5\sigma}\right)^{6}\right% ]\approx-0.0163\varepsilon
  9. r c \displaystyle r_{c}
  10. r c \displaystyle r_{c}
  11. r c \displaystyle r_{c}
  12. V L J \displaystyle V_{LJ}
  13. V L J ( r ) = 4 ε [ ( σ r ) 12 - ( σ r ) 6 ] . \displaystyle V_{LJ}(r)=4\varepsilon\left[\left(\frac{\sigma}{r}\right)^{12}-% \left(\frac{\sigma}{r}\right)^{6}\right].
  14. V L J t r u n c \displaystyle V_{{LJ}_{trunc}}
  15. V L J t r u n c ( r ) := { V L J ( r ) - V L J ( r c ) for r r c 0 for r > r c . \displaystyle V_{{LJ}_{trunc}}(r):=\begin{cases}V_{LJ}(r)-V_{LJ}(r_{c})&\,% \text{for }r\leq r_{c}\\ 0&\,\text{for }r>r_{c}.\end{cases}

Lenstra_elliptic_curve_factorization.html

  1. n n
  2. / n \mathbb{Z}/n\mathbb{Z}
  3. y 2 = x 3 + a x + b ( mod n ) y^{2}=x^{3}+ax+b\;\;(\mathop{{\rm mod}}n)
  4. P ( x 0 , y 0 ) P(x_{0},y_{0})
  5. x 0 , y 0 , a / n x_{0},y_{0},a\in\mathbb{Z}/n\mathbb{Z}
  6. b = y 0 2 - x 0 3 - a x 0 ( mod n ) b=y_{0}^{2}-x_{0}^{3}-ax_{0}\;\;(\mathop{{\rm mod}}n)
  7. \infty
  8. \infty
  9. L p [ 1 2 , 2 ] L_{p}\left[\frac{1}{2},\sqrt{2}\right]
  10. \boxplus
  11. k P = kP=\infty
  12. N p P = N_{p}P=\infty
  13. 3 ( 2 P ) = 4 P 2 P 3(2P)=4P\boxplus 2P
  14. ( / n ) (\mathbb{Z}/n\mathbb{Z})
  15. ( x , y , z ) (x,y,z)
  16. ( x , y , z ) (x,y,z)
  17. ( x , y , z ) (x^{\prime},y^{\prime},z^{\prime})
  18. ( P 2 ) (P^{2})
  19. ( x : y : z ) (x:y:z)
  20. ( 0 : 0 : 0 ) (0:0:0)
  21. ( / n ) (\mathbb{Z}/n\mathbb{Z})
  22. n n
  23. ( 0 : 1 : 0 ) (0:1:0)
  24. n n
  25. E ( Z / n Z ) = { ( x : y : z ) P 2 | y 2 z = x 3 + a x z 2 + b z 3 } E(Z/nZ)=\{(x:y:z)\in P^{2}\ |\ y^{2}z=x^{3}+axz^{2}+bz^{3}\}
  26. x P , y P , a x_{P},y_{P},a
  27. / n \mathbb{Z}/n\mathbb{Z}
  28. a a
  29. b = y P 2 - x P 3 - a x P b=y_{P}^{2}-x_{P}^{3}-ax_{P}
  30. E E
  31. y 2 = x 3 + a x + b y^{2}=x^{3}+ax+b
  32. Z Y 2 = X 3 + a Z 2 X + b Z 3 ZY^{2}=X^{3}+aZ^{2}X+bZ^{3}
  33. P = ( x P : y P : 1 ) P=(x_{P}:y_{P}:1)
  34. B B\in\mathbb{Z}
  35. p p
  36. E E
  37. / p \mathbb{Z}/p\mathbb{Z}
  38. E ( / p ) E(\mathbb{Z}/p\mathbb{Z})
  39. E ( / p ) E(\mathbb{Z}/p\mathbb{Z})
  40. B B
  41. k = lcm ( 1 , , B ) k={\rm lcm}(1,\dots,B)
  42. k P := P + P + + P kP:=P+P+\cdots+P
  43. E ( / n ) E(\mathbb{Z}/n\mathbb{Z})
  44. E ( / n ) E(\mathbb{Z}/n\mathbb{Z})
  45. B B
  46. n n
  47. / n \mathbb{Z}/n\mathbb{Z}
  48. k P = ( 0 : 1 : 0 ) kP=(0:1:0)
  49. E ( / p ) E(\mathbb{Z}/p\mathbb{Z})
  50. p p
  51. n n
  52. n n
  53. k P kP
  54. gcd ( x 1 - x 2 , n ) = 1 \gcd(x_{1}-x_{2},n)=1
  55. P , Q P,Q
  56. ( 0 : 1 : 0 ) (0:1:0)
  57. R = P + Q ; R=P+Q;
  58. P = ( x 1 : y 1 : 1 ) P=(x_{1}:y_{1}:1)
  59. Q = ( x 2 : y 2 : 1 ) Q=(x_{2}:y_{2}:1)
  60. λ = ( y 1 - y 2 ) ( x 1 - x 2 ) - 1 \lambda=(y_{1}-y_{2})(x_{1}-x_{2})^{-1}
  61. x 3 = λ 2 - x 1 - x 2 x_{3}=\lambda^{2}-x_{1}-x_{2}
  62. y 3 = λ ( x 1 - x 3 ) - y 1 y_{3}=\lambda(x_{1}-x_{3})-y_{1}
  63. R = P + Q = ( x 3 : y 3 : 1 ) R=P+Q=(x_{3}:y_{3}:1)
  64. λ \lambda
  65. ( x 1 - x 2 ) - 1 (x_{1}-x_{2})^{-1}
  66. n n
  67. / n \mathbb{Z}/n\mathbb{Z}
  68. / n \mathbb{Z}/n\mathbb{Z}
  69. λ = y 1 - y 2 \lambda^{\prime}=y_{1}-y_{2}
  70. x 3 = λ 2 - x 1 ( x 1 - x 2 ) 2 - x 2 ( x 1 - x 2 ) 2 x_{3}^{\prime}={\lambda^{\prime}}^{2}-x_{1}(x_{1}-x_{2})^{2}-x_{2}(x_{1}-x_{2}% )^{2}
  71. y 3 = λ ( x 1 ( x 1 - x 2 ) 2 - x 3 ) - y 1 ( x 1 - x 2 ) 3 y_{3}^{\prime}=\lambda^{\prime}(x_{1}(x_{1}-x_{2})^{2}-x_{3}^{\prime})-y_{1}(x% _{1}-x_{2})^{3}
  72. R = P + Q = ( x 3 ( x 1 - x 2 ) : y 3 : ( x 1 - x 2 ) 3 ) R=P+Q=(x_{3}^{\prime}(x_{1}-x_{2}):y_{3}^{\prime}:(x_{1}-x_{2})^{3})
  73. Z Z
  74. n n
  75. n n
  76. n n
  77. k k
  78. 2 0 2\neq 0
  79. a , d k { 0 } a,d\in k\setminus\{0\}
  80. a d a\neq d
  81. E E , a , d E_{E,a,d}
  82. a x 2 + y 2 = 1 + d x 2 y 2 . ax^{2}+y^{2}=1+dx^{2}y^{2}.
  83. a = 1 a=1
  84. { ( x , y ) A 2 : a x 2 + y 2 = 1 + d x 2 y 2 } \{(x,y)\in A^{2}:ax^{2}+y^{2}=1+dx^{2}y^{2}\}
  85. ( e , f ) , ( g , h ) ( e h + f g 1 + d e g f h , f h - a e g 1 - d e g f h ) (e,f),(g,h)\mapsto\left(\frac{eh+fg}{1+degfh},\frac{fh-aeg}{1-degfh}\right)
  86. ( e , f ) (e,f)
  87. ( - e , f ) (-e,f)
  88. \mathbb{Q}
  89. / 4 , / 8 , / 12 , / 2 × / 4 \mathbb{Z}/4\mathbb{Z},\mathbb{Z}/8\mathbb{Z},\mathbb{Z}/12\mathbb{Z},\mathbb{% Z}/2\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z}
  90. / 2 × / 8 \mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/8\mathbb{Z}
  91. / 12 \mathbb{Z}/12\mathbb{Z}
  92. / 2 × / 8 \mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/8\mathbb{Z}
  93. / 12 \mathbb{Z}/12\mathbb{Z}
  94. x 2 + y 2 = 1 + d x 2 y 2 x^{2}+y^{2}=1+dx^{2}y^{2}
  95. ( a , b ) (a,b)
  96. b { - 2 , - 1 / 2 , 0 , ± 1 } , a 2 = - ( b 2 + 2 b ) b\notin\{-2,-1/2,0,\pm 1\},a^{2}=-(b^{2}+2b)
  97. d = - ( 2 b + 1 ) / ( a 2 b 2 ) d=-(2b+1)/(a^{2}b^{2})
  98. x 2 + y 2 = 1 + d x 2 y 2 x^{2}+y^{2}=1+dx^{2}y^{2}
  99. ( a , b ) (a,b)
  100. a = u 2 - 1 u 2 + 1 , b = - ( u - 1 ) 2 u 2 + 1 a=\frac{u^{2}-1}{u^{2}+1},b=-\frac{(u-1)^{2}}{u^{2}+1}
  101. d = ( u 2 + 1 ) 3 ( u 2 - 4 u + 1 ) ( u - 1 ) 6 ( u + 1 ) 2 , u { 0 , ± 1 } . d=\frac{(u^{2}+1)^{3}(u^{2}-4u+1)}{(u-1)^{6}(u+1)^{2}},u\notin\{0,\pm 1\}.
  102. / 2 × / 8 \mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/8\mathbb{Z}
  103. / 2 × / 4 \mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z}
  104. p p
  105. s P sP
  106. E ( / p ) E(\mathbb{Z}/p\mathbb{Z})
  107. q q
  108. s P sP
  109. E ( / q ) E(\mathbb{Z}/q\mathbb{Z})
  110. B 1 B_{1}
  111. B 2 B_{2}
  112. B 1 B_{1}
  113. B 2 B_{2}
  114. s P sP
  115. ( l s ) P (ls)P
  116. n n
  117. l l
  118. y 2 = f ( x ) y^{2}=f(x)
  119. f f

Lenz's_law.html

  1. = - Φ t , \mathcal{E}=-\frac{\partial\Phi}{\partial t},
  2. u = 1 2 ε | 𝐄 | 2 , u=\frac{1}{2}\varepsilon|\mathbf{E}|^{2}\,,
  3. δ W = 𝐇 δ 𝐁 . \delta W=\mathbf{H}\cdot\delta\mathbf{B}.

Levi-Civita_connection.html

  1. M n 𝐑 n ( n + 1 ) 2 , M^{n}\subset\mathbf{R}^{\frac{n(n+1)}{2}},
  2. ( M , g ) (M,g)
  3. T M TM
  4. M M
  5. g g
  6. M M
  7. X , Y , Z X,Y,Z
  8. M M
  9. T M TM
  10. X X , Y XX,Y
  11. X X
  12. Y Y
  13. g g
  14. X , Y X,Y
  15. X X
  16. Y Y
  17. p p
  18. g ( X , Y ) g(X,Y)
  19. M M
  20. X f = X i x i f = X i i f , Xf=X^{i}\frac{\partial}{\partial x^{i}}f=X^{i}\partial_{i}f,
  21. g = 0 ∇g=0
  22. X X
  23. Y Y
  24. X X , Y XX,Y
  25. X X
  26. Y Y
  27. g g
  28. X ( g ( Y , Z ) ) + Y ( g ( Z , X ) ) - Z ( g ( Y , X ) ) = g ( X Y + Y X , Z ) + g ( X Z - Z X , Y ) + g ( Y Z - Z Y , X ) . X(g(Y,Z))+Y(g(Z,X))-Z(g(Y,X))=g(\nabla_{X}Y+\nabla_{Y}X,Z)+g(\nabla_{X}Z-% \nabla_{Z}X,Y)+g(\nabla_{Y}Z-\nabla_{Z}Y,X).
  29. 2 g ( X Y , Z ) - g ( [ X , Y ] , Z ) + g ( [ X , Z ] , Y ) + g ( [ Y , Z ] , X ) 2g(\nabla_{X}Y,Z)-g([X,Y],Z)+g([X,Z],Y)+g([Y,Z],X)
  30. g ( X Y , Z ) = 1 2 { X ( g ( Y , Z ) ) + Y ( g ( Z , X ) ) - Z ( g ( X , Y ) ) + g ( [ X , Y ] , Z ) - g ( [ Y , Z ] , X ) - g ( [ X , Z ] , Y ) } . g(\nabla_{X}Y,Z)=\frac{1}{2}\{X(g(Y,Z))+Y(g(Z,X))-Z(g(X,Y))+g([X,Y],Z)-g([Y,Z]% ,X)-g([X,Z],Y)\}.
  31. Z Z
  32. x 1 x n x^{1}\ldots x^{n}
  33. Γ l j k \Gamma^{l}{}_{jk}
  34. Γ l = j k Γ l . k j \Gamma^{l}{}_{jk}=\Gamma^{l}{}_{kj}.
  35. Γ l = j k 1 2 g l r { k g r j + j g r k - r g j k } \Gamma^{l}{}_{jk}=\tfrac{1}{2}g^{lr}\left\{\partial_{k}g_{rj}+\partial_{j}g_{% rk}-\partial_{r}g_{jk}\right\}
  36. g i j g^{ij}
  37. ( g k l ) (g_{kl})
  38. D t V = γ ˙ ( t ) V . D_{t}V=\nabla_{\dot{\gamma}(t)}V.
  39. γ * \gamma^{*}\nabla
  40. γ ˙ ( t ) \dot{\gamma}(t)
  41. γ ˙ ( t ) γ ˙ ( t ) \nabla_{\dot{\gamma}(t)}\dot{\gamma}(t)
  42. γ ˙ \dot{\gamma}
  43. ( γ * ) γ ˙ 0. (\gamma^{*}\nabla)\dot{\gamma}\equiv 0.
  44. , \langle\cdot,\cdot\rangle
  45. Y ( m ) , m = 0 , m 𝐒 2 . \langle Y(m),m\rangle=0,\qquad\forall m\in\mathbf{S}^{2}.
  46. ( X Y ) ( m ) = d m Y ( X ) + X ( m ) , Y ( m ) m \left(\nabla_{X}Y\right)(m)=d_{m}Y(X)+\langle X(m),Y(m)\rangle m
  47. ( X Y ) ( m ) , m = 0 ( 1 ) . \langle\left(\nabla_{X}Y\right)(m),m\rangle=0\qquad(1).
  48. d m f ( X ) = d m Y ( X ) , m + Y ( m ) , X ( m ) = 0. d_{m}f(X)=\langle d_{m}Y(X),m\rangle+\langle Y(m),X(m)\rangle=0.
  49. \Box

Levi-Civita_symbol.html

  1. ε i 1 i 2 i n \varepsilon_{i_{1}i_{2}\cdots i_{n}}
  2. ε i 1 i 2 i n \varepsilon_{i_{1}i_{2}\cdots i_{n}}
  3. ε i p i q = - ε i q i p . \varepsilon_{\cdots i_{p}\cdots i_{q}\cdots}=-\varepsilon_{\cdots i_{q}\cdots i% _{p}\cdots}.
  4. ε i 1 i 2 i n = ( - 1 ) p ε 12 n , \varepsilon_{i_{1}i_{2}\cdots i_{n}}=(-1)^{p}\varepsilon_{12\cdots n},
  5. ε i j = { + 1 if ( i , j ) is ( 1 , 2 ) - 1 if ( i , j ) is ( 2 , 1 ) 0 if i = j \varepsilon_{ij}=\begin{cases}+1&\,\text{if }(i,j)\,\text{ is }(1,2)\\ -1&\,\text{if }(i,j)\,\text{ is }(2,1)\\ \;\;\,0&\,\text{if }i=j\end{cases}
  6. ( ε 11 ε 12 ε 21 ε 22 ) = ( 0 1 - 1 0 ) \begin{pmatrix}\varepsilon_{11}&\varepsilon_{12}\\ \varepsilon_{21}&\varepsilon_{22}\end{pmatrix}=\begin{pmatrix}0&1\\ -1&0\end{pmatrix}
  7. ε i j k = { + 1 if ( i , j , k ) is ( 1 , 2 , 3 ) , ( 2 , 3 , 1 ) or ( 3 , 1 , 2 ) , - 1 if ( i , j , k ) is ( 3 , 2 , 1 ) , ( 1 , 3 , 2 ) or ( 2 , 1 , 3 ) , 0 if i = j or j = k or k = i \varepsilon_{ijk}=\begin{cases}+1&\,\text{if }(i,j,k)\,\text{ is }(1,2,3),(2,3% ,1)\,\text{ or }(3,1,2),\\ -1&\,\text{if }(i,j,k)\,\text{ is }(3,2,1),(1,3,2)\,\text{ or }(2,1,3),\\ \;\;\,0&\,\text{if }i=j\,\text{ or }j=k\,\text{ or }k=i\end{cases}
  8. ε i j k \varepsilon_{ijk}
  9. ε \color B r i c k R e d 1 \color V i o l e t 3 \color O r a n g e 2 = - ε \color B r i c k R e d 1 \color O r a n g e 2 \color V i o l e t 3 = - 1 \varepsilon_{\color{BrickRed}{1}\color{Violet}{3}\color{Orange}{2}}=-% \varepsilon_{\color{BrickRed}{1}\color{Orange}{2}\color{Violet}{3}}=-1
  10. ε \color V i o l e t 3 \color B r i c k R e d 1 \color O r a n g e 2 = - ε \color O r a n g e 2 \color B r i c k R e d 1 \color V i o l e t 3 = - ( - ε \color B r i c k R e d 1 \color O r a n g e 2 \color V i o l e t 3 ) = 1 \varepsilon_{\color{Violet}{3}\color{BrickRed}{1}\color{Orange}{2}}=-% \varepsilon_{\color{Orange}{2}\color{BrickRed}{1}\color{Violet}{3}}=-(-% \varepsilon_{\color{BrickRed}{1}\color{Orange}{2}\color{Violet}{3}})=1
  11. ε \color O r a n g e 2 \color V i o l e t 3 \color B r i c k R e d 1 = - ε \color B r i c k R e d 1 \color V i o l e t 3 \color O r a n g e 2 = - ( - ε \color B r i c k R e d 1 \color O r a n g e 2 \color V i o l e t 3 ) = 1 \varepsilon_{\color{Orange}{2}\color{Violet}{3}\color{BrickRed}{1}}=-% \varepsilon_{\color{BrickRed}{1}\color{Violet}{3}\color{Orange}{2}}=-(-% \varepsilon_{\color{BrickRed}{1}\color{Orange}{2}\color{Violet}{3}})=1
  12. ε \color O r a n g e 2 \color V i o l e t 3 \color O r a n g e 2 = - ε \color O r a n g e 2 \color V i o l e t 3 \color O r a n g e 2 = 0 \varepsilon_{\color{Orange}{2}\color{Violet}{3}\color{Orange}{2}}=-\varepsilon% _{\color{Orange}{2}\color{Violet}{3}\color{Orange}{2}}=0
  13. ε i j k l = { + 1 if ( i , j , k , l ) is an even permutation of ( 1 , 2 , 3 , 4 ) - 1 if ( i , j , k , l ) is an odd permutation of ( 1 , 2 , 3 , 4 ) 0 otherwise \varepsilon_{ijkl}=\begin{cases}+1&\,\text{if }(i,j,k,l)\,\text{ is an even % permutation of }(1,2,3,4)\\ -1&\,\text{if }(i,j,k,l)\,\text{ is an odd permutation of }(1,2,3,4)\\ 0&\,\text{otherwise}\end{cases}
  14. ε \color B r i c k R e d 1 \color R e d V i o l e t 4 \color V i o l e t 3 \color O r a n g e \color O r a n g e 2 = - ε \color B r i c k R e d 1 \color O r a n g e \color O r a n g e 2 \color V i o l e t 3 \color R e d V i o l e t 4 = - 1 \varepsilon_{\color{BrickRed}{1}\color{RedViolet}{4}\color{Violet}{3}\color{% Orange}{\color{Orange}{2}}}=-\varepsilon_{\color{BrickRed}{1}\color{Orange}{% \color{Orange}{2}}\color{Violet}{3}\color{RedViolet}{4}}=-1
  15. ε \color O r a n g e \color O r a n g e 2 \color B r i c k R e d 1 \color V i o l e t 3 \color R e d V i o l e t 4 = - ε \color B r i c k R e d 1 \color O r a n g e \color O r a n g e 2 \color V i o l e t 3 \color R e d V i o l e t 4 = - 1 \varepsilon_{\color{Orange}{\color{Orange}{2}}\color{BrickRed}{1}\color{Violet% }{3}\color{RedViolet}{4}}=-\varepsilon_{\color{BrickRed}{1}\color{Orange}{% \color{Orange}{2}}\color{Violet}{3}\color{RedViolet}{4}}=-1
  16. ε \color R e d V i o l e t 4 \color V i o l e t 3 \color O r a n g e \color O r a n g e 2 \color B r i c k R e d 1 = - ε \color B r i c k R e d 1 \color V i o l e t 3 \color O r a n g e \color O r a n g e 2 \color R e d V i o l e t 4 = - ( - ε \color B r i c k R e d 1 \color O r a n g e \color O r a n g e 2 \color V i o l e t 3 \color R e d V i o l e t 4 ) = 1 \varepsilon_{\color{RedViolet}{4}\color{Violet}{3}\color{Orange}{\color{Orange% }{2}}\color{BrickRed}{1}}=-\varepsilon_{\color{BrickRed}{1}\color{Violet}{3}% \color{Orange}{\color{Orange}{2}}\color{RedViolet}{4}}=-(-\varepsilon_{\color{% BrickRed}{1}\color{Orange}{\color{Orange}{2}}\color{Violet}{3}\color{RedViolet% }{4}})=1
  17. ε \color V i o l e t 3 \color O r a n g e \color O r a n g e 2 \color R e d V i o l e t 4 \color V i o l e t 3 = - ε \color V i o l e t 3 \color O r a n g e \color O r a n g e 2 \color R e d V i o l e t 4 \color V i o l e t 3 = 0 \varepsilon_{\color{Violet}{3}\color{Orange}{\color{Orange}{2}}\color{% RedViolet}{4}\color{Violet}{3}}=-\varepsilon_{\color{Violet}{3}\color{Orange}{% \color{Orange}{2}}\color{RedViolet}{4}\color{Violet}{3}}=0
  18. ε a 1 a 2 a 3 a n = { + 1 if ( a 1 , a 2 , a 3 , , a n ) is an even permutation of ( 1 , 2 , 3 , , n ) - 1 if ( a 1 , a 2 , a 3 , , a n ) is an odd permutation of ( 1 , 2 , 3 , , n ) 0 otherwise \varepsilon_{a_{1}a_{2}a_{3}\ldots a_{n}}=\begin{cases}+1&\,\text{if }(a_{1},a% _{2},a_{3},\ldots,a_{n})\,\text{ is an even permutation of }(1,2,3,\dots,n)\\ -1&\,\text{if }(a_{1},a_{2},a_{3},\ldots,a_{n})\,\text{ is an odd permutation % of }(1,2,3,\dots,n)\\ 0&\,\text{otherwise}\end{cases}
  19. \prod
  20. ε a 1 a 2 a 3 a n \displaystyle\varepsilon_{a_{1}a_{2}a_{3}\ldots a_{n}}
  21. ε i j k = ε i j k . \varepsilon^{ij\dots k}=\varepsilon_{ij\dots k}.
  22. ε i j k ε i m n i = 1 , 2 , 3 ε i j k ε i m n \varepsilon_{ijk}\varepsilon^{imn}\equiv\sum_{i=1,2,3}\varepsilon_{ijk}% \varepsilon^{imn}
  23. ε i j k ε l m n = | δ i l δ i m δ i n δ j l δ j m δ j n δ k l δ k m δ k n | = δ i l ( δ j m δ k n - δ j n δ k m ) - δ i m ( δ j l δ k n - δ j n δ k l ) + δ i n ( δ j l δ k m - δ j m δ k l ) . \begin{aligned}\displaystyle\varepsilon_{ijk}\varepsilon_{lmn}&\displaystyle=% \begin{vmatrix}\delta_{il}&\delta_{im}&\delta_{in}\\ \delta_{jl}&\delta_{jm}&\delta_{jn}\\ \delta_{kl}&\delta_{km}&\delta_{kn}\\ \end{vmatrix}\\ &\displaystyle=\delta_{il}\left(\delta_{jm}\delta_{kn}-\delta_{jn}\delta_{km}% \right)-\delta_{im}\left(\delta_{jl}\delta_{kn}-\delta_{jn}\delta_{kl}\right)+% \delta_{in}\left(\delta_{jl}\delta_{km}-\delta_{jm}\delta_{kl}\right).\end{aligned}
  24. i = 1 3 ε i j k ε i m n = δ j m δ k n - δ j n δ k m \sum_{i=1}^{3}\varepsilon_{ijk}\varepsilon_{imn}=\delta_{jm}\delta_{kn}-\delta% _{jn}\delta_{km}
  25. ε i j k ε i m n = δ j m δ k n - δ j n δ k m . \varepsilon_{ijk}\varepsilon_{imn}=\delta_{jm}\delta_{kn}-\delta_{jn}\delta_{% km}\,.
  26. i = 1 3 j = 1 3 ε i j k ε i j n = 2 δ k n \sum_{i=1}^{3}\sum_{j=1}^{3}\varepsilon_{ijk}\varepsilon_{ijn}=2\delta_{kn}
  27. i , j , k , = 1 n ε i j k ε i j k = n ! \sum_{i,j,k,\dots=1}^{n}\varepsilon_{ijk\dots}\varepsilon_{ijk\dots}=n!
  28. ε i 1 i 2 i n ε j 1 j 2 j n = | δ i 1 j 1 δ i 1 j 2 δ i 1 j n δ i 2 j 1 δ i 2 j 2 δ i 2 j n δ i n j 1 δ i n j 2 δ i n j n | \varepsilon_{i_{1}i_{2}\dots i_{n}}\varepsilon_{j_{1}j_{2}\dots j_{n}}=\begin{% vmatrix}\delta_{i_{1}j_{1}}&\delta_{i_{1}j_{2}}&\dots&\delta_{i_{1}j_{n}}\\ \delta_{i_{2}j_{1}}&\delta_{i_{2}j_{2}}&\dots&\delta_{i_{2}j_{n}}\\ \vdots&\vdots&\ddots&\vdots\\ \delta_{i_{n}j_{1}}&\delta_{i_{n}j_{2}}&\dots&\delta_{i_{n}j_{n}}\\ \end{vmatrix}
  29. ε 12 ε 12 \varepsilon_{12}\varepsilon^{12}
  30. ε i j ε i n = δ i δ j i - n δ i δ j n = i 2 δ j - n δ j = n δ j . n \varepsilon_{ij}\varepsilon^{in}=\delta_{i}{}^{i}\delta_{j}{}^{n}-\delta_{i}{}% ^{n}\delta_{j}{}^{i}=2\delta_{j}{}^{n}-\delta_{j}{}^{n}=\delta_{j}{}^{n}\,.
  31. ε j m n ε i m n = ( ε i m n ) 2 = 1 \varepsilon_{jmn}\varepsilon^{imn}=(\varepsilon^{imn})^{2}=1
  32. ε i j k ε i j k = 1 \varepsilon_{ijk}\varepsilon^{ijk}=1
  33. det ( 𝐀 ) = i = 1 3 j = 1 3 k = 1 3 ε i j k a 1 i a 2 j a 3 k \det(\mathbf{A})=\sum_{i=1}^{3}\sum_{j=1}^{3}\sum_{k=1}^{3}\varepsilon_{ijk}a_% {1i}a_{2j}a_{3k}
  34. det ( 𝐀 ) = ε i 1 i n a 1 i 1 a n i n , \det(\mathbf{A})=\varepsilon_{i_{1}\cdots i_{n}}a_{1i_{1}}\cdots a_{ni_{n}},
  35. det ( 𝐀 ) = 1 n ! ε i 1 i n ε j 1 j n a i 1 j 1 a i n j n , \det(\mathbf{A})=\frac{1}{n!}\varepsilon_{i_{1}\cdots i_{n}}\varepsilon_{j_{1}% \cdots j_{n}}a_{i_{1}j_{1}}\cdots a_{i_{n}j_{n}},
  36. i 1 , i 2 , ε i 1 i n a i 1 j 1 a i n j n = det ( 𝐀 ) ε j 1 j n \sum_{i_{1},i_{2},\dots}\varepsilon_{i_{1}\cdots i_{n}}a_{i_{1}\,j_{1}}\cdots a% _{i_{n}\,j_{n}}=\det(\mathbf{A})\varepsilon_{j_{1}\cdots j_{n}}
  37. 3 \mathbb{R}^{3}
  38. 𝐚 × 𝐛 = | 𝐞 𝟏 𝐞 𝟐 𝐞 𝟑 a 1 a 2 a 3 b 1 b 2 b 3 | = i = 1 3 j = 1 3 k = 1 3 ε i j k 𝐞 i a j b k \mathbf{a\times b}=\begin{vmatrix}\mathbf{e_{1}}&\mathbf{e_{2}}&\mathbf{e_{3}}% \\ a^{1}&a^{2}&a^{3}\\ b^{1}&b^{2}&b^{3}\\ \end{vmatrix}=\sum_{i=1}^{3}\sum_{j=1}^{3}\sum_{k=1}^{3}\varepsilon_{ijk}% \mathbf{e}_{i}a^{j}b^{k}
  39. ( 𝐚 × 𝐛 ) i = j = 1 3 k = 1 3 ε i j k a j b k . (\mathbf{a\times b})_{i}=\sum_{j=1}^{3}\sum_{k=1}^{3}\varepsilon_{ijk}a^{j}b^{% k}.
  40. ( 𝐚 × 𝐛 ) i = ε i j k a j b k . (\mathbf{a\times b})_{i}=\varepsilon_{ijk}a^{j}b^{k}.
  41. ( 𝐚 × 𝐛 ) 1 = a 2 b 3 - a 3 b 2 , (\mathbf{a\times b})_{1}=a^{2}b^{3}-a^{3}b^{2}\,,
  42. ( 𝐚 × 𝐛 ) 2 = a 3 b 1 - a 1 b 3 , (\mathbf{a\times b})_{2}=a^{3}b^{1}-a^{1}b^{3}\,,
  43. ( 𝐚 × 𝐛 ) 3 = a 1 b 2 - a 2 b 1 . (\mathbf{a\times b})_{3}=a^{1}b^{2}-a^{2}b^{1}\,.
  44. 𝐚 × 𝐛 = - 𝐛 × 𝐚 \mathbf{a\times b}=-\mathbf{b\times a}
  45. 𝐚 ( 𝐛 × 𝐜 ) = ε i j k a i b j c k . \mathbf{a}\cdot(\mathbf{b\times c})=\varepsilon_{ijk}a^{i}b^{j}c^{k}.
  46. 𝐚 ( 𝐛 × 𝐜 ) = - 𝐛 ( 𝐚 × 𝐜 ) \mathbf{a}\cdot(\mathbf{b\times c})=-\mathbf{b}\cdot(\mathbf{a\times c})
  47. 3 \mathbb{R}^{3}
  48. ( × 𝐅 ) i ( 𝐱 ) = ε i j k x j F k ( 𝐱 ) , (\nabla\times\mathbf{F})^{i}(\mathbf{x})=\varepsilon^{ijk}\frac{\partial}{% \partial x^{j}}F^{k}(\mathbf{x}),
  49. ε μ 1 μ n = δ 1 n μ 1 μ n \varepsilon^{\mu_{1}\cdots\mu_{n}}=\delta^{\mu_{1}\cdots\mu_{n}}_{\,1\,\cdots% \,n}\,
  50. ε ν 1 ν n = δ ν 1 ν n 1 n . \varepsilon_{\nu_{1}\cdots\nu_{n}}=\delta^{\,1\,\cdots\,n}_{\nu_{1}\cdots\nu_{% n}}\,.
  51. E a 1 a n = | det [ g a b ] | ε a 1 a n , E_{a_{1}\cdots a_{n}}=\sqrt{|\det[g_{ab}]|}\varepsilon_{a_{1}\cdots a_{n}}\,,
  52. g a b g_{ab}
  53. E a 1 a n = ( - 1 ) s E b 1 b n i = 1 n g a i b i , E^{a_{1}\cdots a_{n}}=(-1)^{s}E_{b_{1}\cdots b_{n}}\prod_{i=1}^{n}g^{a_{i}b_{i% }}\,,
  54. E a 1 a n = 1 | det [ g a b ] | ε a 1 a n E^{a_{1}\cdots a_{n}}=\frac{1}{\sqrt{|\det[g_{ab}]|}}\varepsilon^{a_{1}\cdots a% _{n}}\,
  55. E α β γ δ = | det [ g μ ν ] | ε α β γ δ , E_{\alpha\beta\gamma\delta}=\sqrt{|\det[g_{\mu\nu}]|}\varepsilon_{\alpha\beta% \gamma\delta}\,,
  56. E α β γ δ = - g α ζ g β η g γ θ g δ ι E ζ η θ ι . E^{\alpha\beta\gamma\delta}=-g^{\alpha\zeta}g^{\beta\eta}g^{\gamma\theta}g^{% \delta\iota}E_{\zeta\eta\theta\iota}\,.
  57. E α β γ δ E ρ σ μ ν = - g α ζ g β η g γ θ g δ ι δ ζ η θ ι ρ σ μ ν E^{\alpha\beta\gamma\delta}E^{\rho\sigma\mu\nu}=-g^{\alpha\zeta}g^{\beta\eta}g% ^{\gamma\theta}g^{\delta\iota}\delta^{\rho\sigma\mu\nu}_{\zeta\eta\theta\iota}\,
  58. E α β γ δ E ρ σ μ ν = - g α ζ g β η g γ θ g δ ι δ ρ σ μ ν ζ η θ ι E_{\alpha\beta\gamma\delta}E_{\rho\sigma\mu\nu}=-g_{\alpha\zeta}g_{\beta\eta}g% _{\gamma\theta}g_{\delta\iota}\delta^{\zeta\eta\theta\iota}_{\rho\sigma\mu\nu}\,
  59. E α β γ δ E ρ β γ δ = 6 δ ρ α . E^{\alpha\beta\gamma\delta}E_{\rho\beta\gamma\delta}=6\delta^{\alpha}_{\rho}\,.
  60. E α β γ δ E ρ σ γ δ = 2 δ ρ σ α β . E^{\alpha\beta\gamma\delta}E_{\rho\sigma\gamma\delta}=2\delta^{\alpha\beta}_{% \rho\sigma}\,.
  61. E α β γ δ E ρ σ θ δ = δ ρ σ θ α β γ . E^{\alpha\beta\gamma\delta}E_{\rho\sigma\theta\delta}=\delta^{\alpha\beta% \gamma}_{\rho\sigma\theta}\,.

Levinson_recursion.html

  1. y = 𝐌 x . \vec{y}=\mathbf{M}\ \vec{x}.
  2. y \vec{y}
  3. x \vec{x}
  4. 𝐓 n = [ t 0 t - 1 t - 2 t - n + 1 t 1 t 0 t - 1 t - n + 2 t 2 t 1 t 0 t - n + 3 t n - 1 t n - 2 t n - 3 t 0 ] . \mathbf{T}^{n}=\begin{bmatrix}t_{0}&t_{-1}&t_{-2}&\dots&t_{-n+1}\\ t_{1}&t_{0}&t_{-1}&\dots&t_{-n+2}\\ t_{2}&t_{1}&t_{0}&\dots&t_{-n+3}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ t_{n-1}&t_{n-2}&t_{n-3}&\dots&t_{0}\end{bmatrix}.
  5. f n \vec{f}^{n}
  6. 𝐓 n f n = e ^ 1 . \mathbf{T}^{n}\vec{f}^{n}=\hat{e}_{1}.
  7. b n \vec{b}^{n}
  8. 𝐓 n b n = e ^ n . \mathbf{T}^{n}\vec{b}^{n}=\hat{e}_{n}.
  9. 𝐓 n [ f n - 1 0 ] = [ t - n + 1 𝐓 n - 1 t - n + 2 t n - 1 t n - 2 t 0 ] [ f n - 1 0 ] = [ 1 0 0 ϵ f n ] . \mathbf{T}^{n}\begin{bmatrix}\vec{f}^{n-1}\\ 0\\ \end{bmatrix}=\begin{bmatrix}&&&t_{-n+1}\\ &\mathbf{T}^{n-1}&&t_{-n+2}\\ &&&\vdots\\ t_{n-1}&t_{n-2}&\dots&t_{0}\\ \end{bmatrix}\begin{bmatrix}\\ \vec{f}^{n-1}\\ \\ 0\\ \\ \end{bmatrix}=\begin{bmatrix}1\\ 0\\ \vdots\\ 0\\ \epsilon_{f}^{n}\end{bmatrix}.
  10. ϵ f n = i = 1 n - 1 M n i f i n - 1 = i = 1 n - 1 t n - i f i n - 1 . \epsilon_{f}^{n}\ =\ \sum_{i=1}^{n-1}\ M_{ni}\ f_{i}^{n-1}\ =\ \sum_{i=1}^{n-1% }\ t_{n-i}\ f_{i}^{n-1}.
  11. 𝐓 n [ 0 b n - 1 ] = [ t 0 t - n + 2 t - n + 1 t n - 2 𝐓 n - 1 t n - 1 ] [ 0 b n - 1 ] = [ ϵ b n 0 0 1 ] . \mathbf{T}^{n}\begin{bmatrix}0\\ \vec{b}^{n-1}\\ \end{bmatrix}=\begin{bmatrix}t_{0}&\dots&t_{-n+2}&t_{-n+1}\\ \vdots&&&\\ t_{n-2}&&\mathbf{T}^{n-1}&\\ t_{n-1}&&&\end{bmatrix}\begin{bmatrix}\\ 0\\ \\ \vec{b}^{n-1}\\ \\ \end{bmatrix}=\begin{bmatrix}\epsilon_{b}^{n}\\ 0\\ \vdots\\ 0\\ 1\end{bmatrix}.
  12. ϵ b n = i = 2 n M 1 i b i - 1 n - 1 = i = 1 n - 1 t - i b i n - 1 . \epsilon_{b}^{n}\ =\ \sum_{i=2}^{n}\ M_{1i}\ b_{i-1}^{n-1}\ =\ \sum_{i=1}^{n-1% }\ t_{-i}\ b_{i}^{n-1}.
  13. ( α , β ) 𝐓 ( α [ f 0 ] + β [ 0 b ] ) = α [ 1 0 0 ϵ f ] + β [ ϵ b 0 0 1 ] . \forall(\alpha,\beta)\ \mathbf{T}\left(\alpha\begin{bmatrix}\vec{f}\\ \\ 0\\ \end{bmatrix}+\beta\begin{bmatrix}0\\ \\ \vec{b}\end{bmatrix}\right)=\alpha\begin{bmatrix}1\\ 0\\ \vdots\\ 0\\ \epsilon_{f}\\ \end{bmatrix}+\beta\begin{bmatrix}\epsilon_{b}\\ 0\\ \vdots\\ 0\\ 1\end{bmatrix}.
  14. α f n \alpha^{n}_{f}
  15. β f n \beta^{n}_{f}
  16. f n = α f n [ f n - 1 0 ] + β f n [ 0 b n - 1 ] \vec{f}_{n}=\alpha^{n}_{f}\begin{bmatrix}\vec{f}_{n-1}\\ 0\end{bmatrix}+\beta^{n}_{f}\begin{bmatrix}0\\ \vec{b}_{n-1}\end{bmatrix}
  17. α b n \alpha^{n}_{b}
  18. β b n \beta^{n}_{b}
  19. b n = α b n [ f n - 1 0 ] + β b n [ 0 b n - 1 ] . \vec{b}_{n}=\alpha^{n}_{b}\begin{bmatrix}\vec{f}_{n-1}\\ 0\end{bmatrix}+\beta^{n}_{b}\begin{bmatrix}0\\ \vec{b}_{n-1}\end{bmatrix}.
  20. 𝐓 n {\mathbf{T}}^{n}
  21. [ 1 ϵ b n 0 0 0 0 ϵ f n 1 ] [ α f n α b n β f n β b n ] = [ 1 0 0 0 0 0 0 1 ] . \begin{bmatrix}1&\epsilon^{n}_{b}\\ 0&0\\ \vdots&\vdots\\ 0&0\\ \epsilon^{n}_{f}&1\end{bmatrix}\begin{bmatrix}\alpha^{n}_{f}&\alpha^{n}_{b}\\ \beta^{n}_{f}&\beta^{n}_{b}\end{bmatrix}=\begin{bmatrix}1&0\\ 0&0\\ \vdots&\vdots\\ 0&0\\ 0&1\end{bmatrix}.
  22. [ 1 ϵ b n ϵ f n 1 ] [ α f n α b n β f n β b n ] = [ 1 0 0 1 ] . \begin{bmatrix}1&\epsilon^{n}_{b}\\ \epsilon^{n}_{f}&1\end{bmatrix}\begin{bmatrix}\alpha^{n}_{f}&\alpha^{n}_{b}\\ \beta^{n}_{f}&\beta^{n}_{b}\end{bmatrix}=\begin{bmatrix}1&0\\ 0&1\end{bmatrix}.
  23. f n = 1 1 - ϵ b n ϵ f n [ f n - 1 0 ] - ϵ f n 1 - ϵ b n ϵ f n [ 0 b n - 1 ] \vec{f}^{n}={1\over{1-\epsilon_{b}^{n}\epsilon_{f}^{n}}}\begin{bmatrix}\vec{f}% ^{n-1}\\ 0\end{bmatrix}-{\epsilon_{f}^{n}\over{1-\epsilon_{b}^{n}\epsilon_{f}^{n}}}% \begin{bmatrix}0\\ \vec{b}^{n-1}\end{bmatrix}
  24. b n = 1 1 - ϵ b n ϵ f n [ 0 b n - 1 ] - ϵ b n 1 - ϵ b n ϵ f n [ f n - 1 0 ] . \vec{b}^{n}={1\over{1-\epsilon_{b}^{n}\epsilon_{f}^{n}}}\begin{bmatrix}0\\ \vec{b}^{n-1}\end{bmatrix}-{\epsilon_{b}^{n}\over{1-\epsilon_{b}^{n}\epsilon_{% f}^{n}}}\begin{bmatrix}\vec{f}^{n-1}\\ 0\end{bmatrix}.
  25. f 1 = b 1 = [ 1 M 11 ] = [ 1 t 0 ] . \vec{f}^{1}=\vec{b}^{1}=\begin{bmatrix}{1\over M_{11}}\end{bmatrix}=\begin{% bmatrix}{1\over t_{0}}\end{bmatrix}.
  26. y = 𝐌 x . \vec{y}=\mathbf{M}\ \vec{x}.
  27. x \vec{x}
  28. x n \vec{x}^{n}
  29. x N = x \vec{x}^{N}=\vec{x}
  30. 𝐓 n - 1 [ x 1 n - 1 x 2 n - 1 x n - 1 n - 1 ] = [ y 1 y 2 y n - 1 ] . \mathbf{T}^{n-1}\begin{bmatrix}x_{1}^{n-1}\\ x_{2}^{n-1}\\ \vdots\\ x_{n-1}^{n-1}\\ \end{bmatrix}=\begin{bmatrix}y_{1}\\ y_{2}\\ \vdots\\ y_{n-1}\end{bmatrix}.
  31. 𝐓 n [ x 1 n - 1 x 2 n - 1 x n - 1 n - 1 0 ] = [ y 1 y 2 y n - 1 ϵ x n - 1 ] . \mathbf{T}^{n}\begin{bmatrix}x_{1}^{n-1}\\ x_{2}^{n-1}\\ \vdots\\ x_{n-1}^{n-1}\\ 0\end{bmatrix}=\begin{bmatrix}y_{1}\\ y_{2}\\ \vdots\\ y_{n-1}\\ \epsilon_{x}^{n-1}\end{bmatrix}.
  32. 𝐓 n ( [ x 1 n - 1 x 2 n - 1 x n - 1 n - 1 0 ] + ( y n - ϵ x n - 1 ) b n ) = [ y 1 y 2 y n - 1 y n ] . \mathbf{T}^{n}\left(\begin{bmatrix}x_{1}^{n-1}\\ x_{2}^{n-1}\\ \vdots\\ x_{n-1}^{n-1}\\ 0\\ \end{bmatrix}+(y_{n}-\epsilon_{x}^{n-1})\ \vec{b}^{n}\right)=\begin{bmatrix}y_% {1}\\ y_{2}\\ \vdots\\ y_{n-1}\\ y_{n}\end{bmatrix}.
  33. x \vec{x}

Liber_Abaci.html

  1. 2 1 3 \scriptstyle 2\,\frac{1}{3}
  2. 1 3 2 \scriptstyle\frac{1}{3}\,2
  3. b a d c = a c + b c d \scriptstyle\frac{b\,\,a}{d\,\,c}=\frac{a}{c}+\frac{b}{cd}
  4. c b a f e d = a d + b d e + c d e f \scriptstyle\frac{c\,\,b\,\,a}{f\,\,e\,\,d}=\frac{a}{d}+\frac{b}{de}+\frac{c}{def}
  5. 1  2  4 2  3  5 \scriptstyle\frac{1\,\,2\,\,4}{2\,\,3\,\,5}
  6. 4 5 + 2 3 × 5 + 1 2 × 3 × 5 \scriptstyle\frac{4}{5}+\frac{2}{3\times 5}+\frac{1}{2\times 3\times 5}
  7. 7 3 4 \scriptstyle 7\frac{3}{4}
  8. 3  7  2 4  12  3 5 \scriptstyle\frac{3\ \,7\,\,2}{4\,\,12\,\,3}\,5
  9. 1 4 1 3 2 \scriptstyle\frac{1}{4}\,\frac{1}{3}\,2
  10. 2 7 12 \scriptstyle 2\,\frac{7}{12}
  11. 31 12 \scriptstyle\frac{31}{12}

Lie_algebroid.html

  1. ( E , [ , ] , ρ ) (E,[\cdot,\cdot],\rho)
  2. E E
  3. M M
  4. [ , ] [\cdot,\cdot]
  5. Γ ( E ) \Gamma(E)
  6. ρ : E T M \rho:E\rightarrow TM
  7. T M TM
  8. M M
  9. [ X , f Y ] = ρ ( X ) f Y + f [ X , Y ] [X,fY]=\rho(X)f\cdot Y+f[X,Y]
  10. X , Y Γ ( E ) , f C ( M ) X,Y\in\Gamma(E),f\in C^{\infty}(M)
  11. ρ ( X ) f \rho(X)f
  12. f f
  13. ρ ( X ) \rho(X)
  14. ρ ( [ X , Y ] ) = [ ρ ( X ) , ρ ( Y ) ] \rho([X,Y])=[\rho(X),\rho(Y)]
  15. X , Y Γ ( E ) X,Y\in\Gamma(E)
  16. T M TM
  17. M M
  18. T M TM
  19. T M TM
  20. M M
  21. 0 P × G 𝔤 T P / G 𝜌 T M 0. 0\to P\times_{G}\mathfrak{g}\to TP/G\xrightarrow{\rho}TM\to 0.
  22. e : M G e:M\to G
  23. t : G M t:G\to M
  24. T t G = p M T ( t - 1 ( p ) ) T G T^{t}G=\bigcup_{p\in M}T(t^{-1}(p))\subset TG
  25. A := e * T t G A:=e^{*}T^{t}G
  26. T s : e * T t G T M Ts:e^{*}T^{t}G\rightarrow TM
  27. G := M × M G:=M\times M
  28. t : G M : ( p , q ) p t:G\to M:(p,q)\mapsto p
  29. e : M G : p ( p , p ) e:M\to G:p\mapsto(p,p)
  30. p × M p\times M
  31. T t G = p M p × T M T M × T M T^{t}G=\bigcup_{p\in M}p\times TM\subset TM\times TM
  32. A := e * T t G = p M T p M = T M A:=e^{*}T^{t}G=\bigcup_{p\in M}T_{p}M=TM
  33. X ~ ( p , q ) = 0 X ( q ) \tilde{X}(p,q)=0\oplus X(q)
  34. f ~ ( p , q ) = f ( q ) \tilde{f}(p,q)=f(q)
  35. i * i_{*}
  36. i : G G i:G\to G

Lie_derivative.html

  1. f f\,
  2. X X\,
  3. X ( f ) X(f)\,
  4. X ( Y ) = [ X , Y ] \mathcal{L}_{X}(Y)=[X,Y]
  5. [ X , Y ] = [ X , Y ] . \mathcal{L}_{[X,Y]}=[\mathcal{L}_{X},\mathcal{L}_{Y}].
  6. X f \mathcal{L}_{X}f
  7. X X
  8. ( X f ) ( p ) X p ( f ) ( X f ) ( p ) . (\mathcal{L}_{\!X}f)(p)\triangleq X_{p}(f)\triangleq(Xf)(p).
  9. ( X f ) ( p ) d f p ( X p ) . (\mathcal{L}_{\!X}f)(p)\triangleq\operatorname{d}f_{p}\,(X_{p}).
  10. X = X a a X=X^{a}\partial_{a}
  11. a = x a \partial_{a}=\frac{\partial}{\partial x^{a}}
  12. T M TM
  13. ( X f ) ( p ) = X a ( p ) ( a f ) ( p ) . (\mathcal{L}_{\!X}f)(p)=X^{a}(p)(\partial_{a}f)(p).
  14. d f : M T * M \operatorname{d}f:M\to T^{*}M
  15. d f a f d x a \operatorname{d}f\triangleq\partial_{a}f\operatorname{d}x^{a}
  16. ( X f ) ( p ) = d f p ( X p ) = X a ( p ) ( b f ) ( p ) d x b ( a ) = X a ( p ) ( a f ) ( p ) (\mathcal{L}_{\!X}f)(p)=\operatorname{d}f_{p}\,(X_{p})=X^{a}(p)(\partial_{b}f)% (p)\,dx^{b}(\partial_{a})=X^{a}(p)(\partial_{a}f)(p)
  17. ( X f ) ( p ) d d t f ( γ ( t ) ) | t = 0 \left.(\mathcal{L}_{\!X}f)(p)\triangleq\frac{\operatorname{d}}{\operatorname{d% }t}f(\gamma(t))\right|_{t=0}
  18. γ ( t ) \gamma(t)
  19. γ ( 0 ) = p \gamma(0)=p
  20. γ ( 0 ) = X p \gamma^{\prime}(0)=X_{p}
  21. X X
  22. d d t γ ( t ) = X γ ( t ) \frac{d}{dt}\gamma(t)=X_{\gamma(t)}
  23. [ X , Y ] [X,Y]
  24. X Y = [ X , Y ] . \mathcal{L}_{X}Y=[X,Y].\,
  25. Φ t X \Phi^{X}_{t}
  26. ( X Y ) x := lim t 0 ( d Φ - t X ) Y Φ t X ( x ) - Y x t = d d t | t = 0 ( d Φ - t X ) Y Φ t X ( x ) (\mathcal{L}_{X}Y)_{x}:=\lim_{t\to 0}\frac{(\mathrm{d}\Phi^{X}_{-t})Y_{\Phi^{X% }_{t}(x)}-Y_{x}}{t}=\left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_{t=0}(\mathrm{% d}\Phi^{X}_{-t})Y_{\Phi^{X}_{t}(x)}
  27. X Y := 1 2 d 2 dt 2 | t = 0 Φ - t Y Φ - t X Φ t Y Φ t X = d d t | t = 0 Φ - t Y Φ - t X Φ t Y Φ t X . \mathcal{L}_{X}Y:=\left.\frac{1}{2}\frac{\mathrm{d}^{2}}{\mathrm{dt}^{2}}% \right|_{t=0}\Phi^{Y}_{-t}\circ\Phi^{X}_{-t}\circ\Phi^{Y}_{t}\circ\Phi^{X}_{t}% =\left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_{t=0}\Phi^{Y}_{-\sqrt{t}}\circ% \Phi^{X}_{-\sqrt{t}}\circ\Phi^{Y}_{\sqrt{t}}\circ\Phi^{X}_{\sqrt{t}}.\,
  28. ω Λ k + 1 ( M ) \omega\in\Lambda^{k+1}(M)
  29. i X ω i_{X}\omega
  30. ( i X ω ) ( X 1 , , X k ) = ω ( X , X 1 , , X k ) (i_{X}\omega)(X_{1},\ldots,X_{k})=\omega(X,X_{1},\ldots,X_{k})\,
  31. i X ω i_{X}\omega
  32. i X : Λ k + 1 ( M ) Λ k ( M ) i_{X}:\Lambda^{k+1}(M)\rightarrow\Lambda^{k}(M)\,
  33. i X i_{X}
  34. \wedge
  35. i X i_{X}
  36. i X ( ω η ) = ( i X ω ) η + ( - 1 ) k ω ( i X η ) i_{X}(\omega\wedge\eta)=(i_{X}\omega)\wedge\eta+(-1)^{k}\omega\wedge(i_{X}\eta)
  37. ω Λ k ( M ) \omega\in\Lambda^{k}(M)
  38. f Λ 0 ( M ) f\in\Lambda^{0}(M)
  39. i f X ω = f i X ω i_{fX}\omega=f\,i_{X}\omega
  40. f X fX
  41. X f = i X d f \mathcal{L}_{X}f=i_{X}\,df
  42. X ω = i X d ω + d ( i X ω ) . \mathcal{L}_{X}\omega=i_{X}d\omega+d(i_{X}\omega).
  43. d X ω = X ( d ω ) . d\mathcal{L}_{X}\omega=\mathcal{L}_{X}(d\omega).
  44. f X ω = f X ω + d f i X ω \mathcal{L}_{fX}\omega=f\mathcal{L}_{X}\omega+df\wedge i_{X}\omega
  45. ( M ) \mathcal{F}(M)
  46. X : ( M ) ( M ) \mathcal{L}_{X}:\mathcal{F}(M)\rightarrow\mathcal{F}(M)
  47. ( M ) \mathcal{F}(M)
  48. X \mathcal{L}_{X}
  49. X ( f g ) = ( X f ) g + f X g . \mathcal{L}_{X}(fg)=(\mathcal{L}_{X}f)g+f\mathcal{L}_{X}g.
  50. ( M ) × 𝒳 ( M ) \mathcal{F}(M)\times\mathcal{X}(M)
  51. 𝒳 ( M ) \mathcal{X}(M)
  52. X ( f Y ) = ( X f ) Y + f X Y \mathcal{L}_{X}(fY)=(\mathcal{L}_{X}f)Y+f\mathcal{L}_{X}Y
  53. X ( f Y ) = ( X f ) Y + f X Y \mathcal{L}_{X}(f\otimes Y)=(\mathcal{L}_{X}f)\otimes Y+f\otimes\mathcal{L}_{X}Y
  54. \otimes
  55. X [ Y , Z ] = [ X Y , Z ] + [ Y , X Z ] \mathcal{L}_{X}[Y,Z]=[\mathcal{L}_{X}Y,Z]+[Y,\mathcal{L}_{X}Z]
  56. X ( α β ) = ( X α ) β + α ( X β ) \mathcal{L}_{X}(\alpha\wedge\beta)=(\mathcal{L}_{X}\alpha)\wedge\beta+\alpha% \wedge(\mathcal{L}_{X}\beta)
  57. [ X , Y ] α := X Y α - Y X α = [ X , Y ] α [\mathcal{L}_{X},\mathcal{L}_{Y}]\alpha:=\mathcal{L}_{X}\mathcal{L}_{Y}\alpha-% \mathcal{L}_{Y}\mathcal{L}_{X}\alpha=\mathcal{L}_{[X,Y]}\alpha
  58. [ X , i Y ] α = [ i X , Y ] α = i [ X , Y ] α , [\mathcal{L}_{X},i_{Y}]\alpha=[i_{X},\mathcal{L}_{Y}]\alpha=i_{[X,Y]}\alpha,
  59. ( q , r ) (q,r)
  60. ( Y T ) p = d d t | t = 0 ( ( φ - t ) * T φ t ( p ) ) = d d t | t = 0 ( ( φ t ) * T ) p . (\mathcal{L}_{Y}T)_{p}=\left.\frac{d}{dt}\right|_{t=0}\left((\varphi_{-t})_{*}% T_{\varphi_{t}(p)}\right)=\left.\frac{d}{dt}\right|_{t=0}\left((\varphi_{t})^{% *}T\right)_{p}.
  61. ( φ t ) * (\varphi_{t})_{*}
  62. ( φ t ) * (\varphi_{t})^{*}
  63. T T
  64. Y T \mathcal{L}_{Y}T
  65. T T
  66. T T
  67. Y f = Y ( f ) \mathcal{L}_{Y}f=Y(f)
  68. Y ( S T ) = ( Y S ) T + S ( Y T ) . \mathcal{L}_{Y}(S\otimes T)=(\mathcal{L}_{Y}S)\otimes T+S\otimes(\mathcal{L}_{% Y}T).
  69. X ( T ( Y 1 , , Y n ) ) = ( X T ) ( Y 1 , , Y n ) + T ( ( X Y 1 ) , , Y n ) + + T ( Y 1 , , ( X Y n ) ) \mathcal{L}_{X}(T(Y_{1},\ldots,Y_{n}))=(\mathcal{L}_{X}T)(Y_{1},\ldots,Y_{n})+% T((\mathcal{L}_{X}Y_{1}),\ldots,Y_{n})+\cdots+T(Y_{1},\ldots,(\mathcal{L}_{X}Y% _{n}))
  70. [ X , d ] = 0 [\mathcal{L}_{X},d]=0
  71. d f ( Y ) = Y ( f ) df(Y)=Y(f)
  72. Y X = [ Y , X ] . \mathcal{L}_{Y}X=[Y,X].
  73. Y α = i Y d α + d i Y α . \mathcal{L}_{Y}\alpha=i_{Y}d\alpha+di_{Y}\alpha.
  74. ( Y T ) ( α 1 , α 2 , , X 1 , X 2 , ) = Y ( T ( α 1 , α 2 , , X 1 , X 2 , ) ) (\mathcal{L}_{Y}T)(\alpha_{1},\alpha_{2},\ldots,X_{1},X_{2},\ldots)=Y(T(\alpha% _{1},\alpha_{2},\ldots,X_{1},X_{2},\ldots))
  75. - T ( Y α 1 , α 2 , , X 1 , X 2 , ) - T ( α 1 , Y α 2 , , X 1 , X 2 , ) - -T(\mathcal{L}_{Y}\alpha_{1},\alpha_{2},\ldots,X_{1},X_{2},\ldots)-T(\alpha_{1% },\mathcal{L}_{Y}\alpha_{2},\ldots,X_{1},X_{2},\ldots)-\ldots
  76. - T ( α 1 , α 2 , , Y X 1 , X 2 , ) - T ( α 1 , α 2 , , X 1 , Y X 2 , ) - -T(\alpha_{1},\alpha_{2},\ldots,\mathcal{L}_{Y}X_{1},X_{2},\ldots)-T(\alpha_{1% },\alpha_{2},\ldots,X_{1},\mathcal{L}_{Y}X_{2},\ldots)-\ldots
  77. T T
  78. X X
  79. ( X T ) a 1 a r = b 1 b s X c ( c T a 1 a r ) b 1 b s - ( c X a 1 ) T c a 2 a r - b 1 b s - ( c X a r ) T a 1 a r - 1 c b 1 b s + ( b 1 X c ) T a 1 a r + c b 2 b s + ( b s X c ) T a 1 a r b 1 b s - 1 c \begin{aligned}\displaystyle(\mathcal{L}_{X}T)^{a_{1}\ldots a_{r}}{}_{b_{1}% \ldots b_{s}}=&\displaystyle X^{c}(\partial_{c}T^{a_{1}\ldots a_{r}}{}_{b_{1}% \ldots b_{s}})\\ &\displaystyle-(\partial_{c}X^{a_{1}})T^{ca_{2}\ldots a_{r}}{}_{b_{1}\ldots b_% {s}}-\ldots-(\partial_{c}X^{a_{r}})T^{a_{1}\ldots a_{r-1}c}{}_{b_{1}\ldots b_{% s}}\\ &\displaystyle+(\partial_{b_{1}}X^{c})T^{a_{1}\ldots a_{r}}{}_{cb_{2}\ldots b_% {s}}+\ldots+(\partial_{b_{s}}X^{c})T^{a_{1}\ldots a_{r}}{}_{b_{1}\ldots b_{s-1% }c}\end{aligned}
  80. a = x a \partial_{a}=\frac{\partial}{\partial x^{a}}
  81. x a x^{a}
  82. a \partial_{a}
  83. a \nabla_{a}
  84. ( X T ) a 1 a r a 1 b 1 b s a r d x b 1 d x b s (\mathcal{L}_{X}T)^{a_{1}\ldots a_{r}}{}_{b_{1}\ldots b_{s}}\partial_{a_{1}}% \otimes\cdots\otimes\partial_{a_{r}}\otimes dx^{b_{1}}\otimes\cdots\otimes dx^% {b_{s}}
  85. ( X T ) a 1 a r = b 1 b s X c ( c T a 1 a r ) b 1 b s - ( c X a 1 ) T c a 2 a r - b 1 b s - ( c X a r ) T a 1 a r - 1 c + b 1 b s (\mathcal{L}_{X}T)^{a_{1}\ldots a_{r}}{}_{b_{1}\ldots b_{s}}=X^{c}(\partial_{c% }T^{a_{1}\ldots a_{r}}{}_{b_{1}\ldots b_{s}})-(\partial_{c}X^{a_{1}})T^{ca_{2}% \ldots a_{r}}{}_{b_{1}\ldots b_{s}}-\ldots-(\partial_{c}X^{a_{r}})T^{a_{1}% \ldots a_{r-1}c}{}_{b_{1}\ldots b_{s}}+
  86. + ( b 1 X c ) T a 1 a r + c b 2 b s + ( b s X c ) T a 1 a r + b 1 b s - 1 c w ( c X c ) T a 1 a r b 1 b s +(\partial_{b_{1}}X^{c})T^{a_{1}\ldots a_{r}}{}_{cb_{2}\ldots b_{s}}+\ldots+(% \partial_{b_{s}}X^{c})T^{a_{1}\ldots a_{r}}{}_{b_{1}\ldots b_{s-1}c}+w(% \partial_{c}X^{c})T^{a_{1}\ldots a_{r}}{}_{b_{1}\ldots b_{s}}
  87. ϕ ( x c ) ( M ) \phi(x^{c})\in\mathcal{F}(M)
  88. ( X ϕ ) = X a a ϕ (\mathcal{L}_{X}\phi)=X^{a}\partial_{a}\phi
  89. A = A a ( x b ) d x a A=A_{a}(x^{b})dx^{a}
  90. ( X A ) a = X b b A a + A b a X b (\mathcal{L}_{X}A)_{a}=X^{b}\partial_{b}A_{a}+A_{b}\partial_{a}X^{b}
  91. g = g a b ( x c ) d x a d x b g=g_{ab}(x^{c})dx^{a}\otimes dx^{b}
  92. ( X g ) a b = X c c g a b + g c b a X c + g c a b X c (\mathcal{L}_{X}g)_{ab}=X^{c}\partial_{c}g_{ab}+g_{cb}\partial_{a}X^{c}+g_{ca}% \partial_{b}X^{c}
  93. ( M , g ) (M,g)
  94. ψ \psi
  95. X ψ := X a a ψ - 1 4 a X b γ a γ b ψ , \mathcal{L}_{X}\psi:=X^{a}\nabla_{a}\psi-\frac{1}{4}\nabla_{a}X_{b}\gamma^{a}% \,\gamma^{b}\psi\,,
  96. a X b = [ a X b ] \nabla_{a}X_{b}=\nabla_{[a}X_{b]}
  97. X = X a a X=X^{a}\partial_{a}
  98. γ a \gamma^{a}
  99. X X
  100. a X b \nabla_{a}X_{b}
  101. X ψ := X a a ψ - 1 8 [ a X b ] [ γ a , γ b ] ψ = X ψ - 1 4 ( d X ) ψ , \mathcal{L}_{X}\psi:=X^{a}\nabla_{a}\psi-\frac{1}{8}\nabla_{[a}X_{b]}[\gamma^{% a},\gamma^{b}]\psi\,=\nabla_{X}\psi-\frac{1}{4}(dX^{\flat})\cdot\psi\,,
  102. [ γ a , γ b ] = γ a γ b - γ b γ a [\gamma^{a},\gamma^{b}]=\gamma^{a}\gamma^{b}-\gamma^{b}\gamma^{a}
  103. d d
  104. X = g ( X , - ) X^{\flat}=g(X,-)
  105. X X
  106. \cdot
  107. K α = [ d , i K ] α = d i K α - ( - 1 ) k - 1 i K d α . \mathcal{L}_{K}\alpha=[d,i_{K}]\alpha=di_{K}\alpha-(-1)^{k-1}i_{K}\,d\alpha.
  108. δ A \delta^{\ast}A
  109. A A\,
  110. X X\,
  111. δ A \delta^{\ast}A
  112. - X ( A ) -\mathcal{L}_{X}(A)\,

Lie_groupoid.html

  1. O b Ob
  2. M o r Mor
  3. s , t : M o r O b s,t:Mor\to Ob
  4. M M
  5. M M
  6. M × M M\times M
  7. G G
  8. M M
  9. g G , x , y M g\in G,x,y\in M
  10. g x = y gx=y
  11. P M P\to M
  12. P × P / G P\times P/G
  13. { U α } \{U_{\alpha}\}
  14. G 0 := α U α G_{0}:=\bigsqcup_{\alpha}U_{\alpha}
  15. p : G 0 M p:G_{0}\to M
  16. G 1 := α , β U α β G_{1}:=\bigsqcup_{\alpha,\beta}U_{\alpha\beta}
  17. U α β = U α U β M U_{\alpha\beta}=U_{\alpha}\cap U_{\beta}\subset M
  18. s : U α β U α s:U_{\alpha\beta}\to U_{\alpha}
  19. t : U α β U β t:U_{\alpha\beta}\to U_{\beta}
  20. U α β U_{\alpha\beta}
  21. U α β U_{\alpha\beta}
  22. U β γ U_{\beta\gamma}
  23. U α γ U_{\alpha\gamma}
  24. M M M\Rightarrow M
  25. G 1 G 0 G_{1}\Rightarrow G_{0}
  26. H 1 H 0 H_{1}\Rightarrow H_{0}
  27. K 1 K 0 K_{1}\Rightarrow K_{0}
  28. G 0 / G 1 = H 0 / H 1 G_{0}/G_{1}=H_{0}/H_{1}
  29. G p H q G_{p}\cong H_{q}
  30. p G 0 p\in G_{0}
  31. q H 0 q\in H_{0}

Lie_superalgebra.html

  1. [ x , y ] = - ( - 1 ) | x | | y | [ y , x ] . [x,y]=-(-1)^{|x||y|}[y,x].
  2. [ x , [ y , z ] ] = [ [ x , y ] , z ] + ( - 1 ) | x | | y | [ y , [ x , z ] ] [x,[y,z]]=[[x,y],z]+(-1)^{|x||y|}[y,[x,z]]
  3. [ x , x ] = 0 [x,x]=0
  4. [ [ x , x ] , x ] = 0 [[x,x],x]=0
  5. Z 2 Z_{2}
  6. { , } : L 1 L 1 L 0 \{\cdot,\cdot\}:L_{1}\otimes L_{1}\rightarrow L_{0}
  7. { x , y } [ z ] + { y , z } [ x ] + { z , x } [ y ] = 0. \left\{x,y\right\}[z]+\left\{y,z\right\}[x]+\left\{z,x\right\}[y]=0.
  8. [ x , y ] = x y - ( - 1 ) | x | | y | y x [x,y]=xy-(-1)^{|x||y|}yx
  9. z . z ¯ + i w . w ¯ z.\overline{z}+iw.\overline{w}
  10. S U ( m / n ) = O S p ( 2 m / 2 n ) O S p ( 2 n / 2 m ) SU(m/n)=OSp(2m/2n)\cap OSp(2n/2m)
  11. x . x + y . z - z . y x.x+y.z-z.y
  12. α \alpha
  13. α \alpha
  14. A μ A μ + B μ B μ + C μ C μ + ψ α β γ ψ α β γ ε α α ε β β ε γ γ A_{\mu}A_{\mu}+B_{\mu}B_{\mu}+C_{\mu}C_{\mu}+\psi^{\alpha\beta\gamma}\psi^{% \alpha^{\prime}\beta^{\prime}\gamma^{\prime}}\varepsilon_{\alpha\alpha^{\prime% }}\varepsilon_{\beta\beta^{\prime}}\varepsilon_{\gamma\gamma^{\prime}}
  15. A { 1 A 2 A 3 } + B { 1 B 2 B 3 } + C { 1 C 2 C 3 } + A μ Γ μ α α ψ ψ + B μ Γ μ β β ψ ψ + C μ Γ μ γ γ ψ ψ A_{\{1}A_{2}A_{3\}}+B_{\{1}B_{2}B_{3\}}+C_{\{1}C_{2}C_{3\}}+A_{\mu}\Gamma^{% \alpha\alpha^{\prime}}_{\mu}\psi\psi+B_{\mu}\Gamma^{\beta\beta^{\prime}}_{\mu}% \psi\psi+C_{\mu}\Gamma^{\gamma\gamma^{\prime}}_{\mu}\psi\psi
  16. γ \gamma
  17. B μ ν + B ν μ = 0 B_{\mu\nu}+B_{\nu\mu}=0
  18. A μ A μ + B μ ν B μ ν + ψ { 1 α ψ 2 } α A_{\mu}A_{\mu}+B_{\mu\nu}B_{\mu\nu}+\psi_{\{1}^{\alpha}\psi_{2\}}^{\alpha}
  19. A { 1 A 2 A 3 } + B { μ ν B ν τ B τ μ } + B μ ν σ μ ν α β ψ k α ψ k β + A μ Γ μ α β ψ α k ψ β k + ( s y m . ) A_{\{1}A_{2}A_{3\}}+B_{\{\mu\nu}B_{\nu\tau}B_{\tau\mu\}}+B_{\mu\nu}\sigma_{\mu% \nu}^{\alpha\beta}\psi^{\alpha}_{k}\psi^{\beta}_{k}+A_{\mu}\Gamma_{\mu}^{% \alpha\beta}\psi^{k}_{\alpha}\psi^{k}_{\beta}+(sym.)
  20. f μ ν τ σ ν τ γ μ f^{\mu\nu\tau}\sigma_{\nu\tau}\equiv\gamma_{\mu}
  21. A μ A μ + C α μ C α μ + ψ { 1 μ ψ 2 } ν A_{\mu}A_{\mu}+C^{\mu}_{\alpha}C^{\mu}_{\alpha}+\psi_{\{1}^{\mu}\psi_{2\}}^{\nu}
  22. [ , ] ( i d + τ A , A ) = 0 [\cdot,\cdot]\circ(id+\tau_{A,A})=0
  23. [ , ] ( [ , ] i d ) ( i d + σ + σ 2 ) = 0 [\cdot,\cdot]\circ([\cdot,\cdot]\otimes id)\circ(id+\sigma+\sigma^{2})=0
  24. ( i d τ A , A ) ( τ A , A i d ) (id\otimes\tau_{A,A})\circ(\tau_{A,A}\otimes id)

Lift-induced_drag.html

  1. D i = 1 2 ρ V 2 S C D i = 1 2 ρ 0 V e 2 S C D i D_{i}=\frac{1}{2}\rho V^{2}SC_{Di}=\frac{1}{2}\rho_{0}V_{e}^{2}SC_{Di}
  2. C D i = C L 2 π e A R C_{Di}=\frac{C_{L}^{2}}{\pi eAR}
  3. C L = L 1 2 ρ 0 V e 2 S C_{L}=\frac{L}{\frac{1}{2}\rho_{0}V_{e}^{2}S}
  4. C D i = L 2 1 4 ρ 0 2 V e 4 S 2 π e A R C_{Di}=\frac{L^{2}}{\frac{1}{4}\rho_{0}^{2}V_{e}^{4}S^{2}\pi eAR}
  5. D i = L 2 1 2 ρ 0 V e 2 S π e A R D_{i}=\frac{L^{2}}{\frac{1}{2}\rho_{0}V_{e}^{2}S\pi eAR}
  6. A R AR\,
  7. C D i C_{Di}\,
  8. C L C_{L}\,
  9. D i D_{i}\,
  10. e e\,
  11. L L\,
  12. S S\,
  13. V V\,
  14. V e V_{e}\,
  15. ρ \rho\,
  16. ρ 0 \rho_{0}\,

Lift-to-drag_ratio.html

  1. ( L / D ) m a x = 1 2 π A ϵ C D , 0 (L/D)_{max}=\frac{1}{2}\sqrt{\frac{\pi A\epsilon}{C_{D,0}}}
  2. ϵ \epsilon
  3. C D , 0 C_{D,0}
  4. C D , 0 = C f e S w e t S r e f C_{D,0}=C_{fe}\frac{S_{wet}}{S_{ref}}
  5. C f e C_{fe}
  6. S w e t S_{wet}
  7. S r e f S_{ref}
  8. b 2 / S r e f b^{2}/S_{ref}
  9. ( L / D ) m a x = 1 2 π ϵ C f e b 2 S w e t (L/D)_{max}=\frac{1}{2}\sqrt{\frac{\pi\epsilon}{C_{fe}}\frac{b^{2}}{S_{wet}}}
  10. b 2 / S w e t b^{2}/S_{wet}
  11. L / D m a x = 4 ( M + 3 ) M L/D_{max}=\frac{4(M+3)}{M}

Lift_coefficient.html

  1. C L = L 1 2 ρ v 2 S = 2 L ρ v 2 S = L q S C_{\mathrm{L}}={\frac{L}{\frac{1}{2}\rho v^{2}S}}={\frac{2L}{\rho v^{2}S}}=% \frac{L}{qS}
  2. L L\,
  3. ρ \rho\,
  4. v v\,
  5. S S\,
  6. q q\,
  7. c l c\text{l}
  8. l l
  9. c l = l 1 2 ρ v 2 c , c\text{l}=\frac{l}{\frac{1}{2}\rho v^{2}c},
  10. c c\,

Light_cone.html

  1. p p
  2. t 0 t_{0}
  3. p p
  4. p p
  5. p p
  6. p p
  7. E E
  8. E E
  9. E E
  10. E E
  11. E E
  12. E E
  13. E E
  14. E E
  15. E E

Light_meter.html

  1. N 2 t = L S K \frac{N^{2}}{t}=\frac{LS}{K}
  2. N N
  3. t t
  4. L L
  5. S S
  6. K K
  7. N 2 t = E S C \frac{N^{2}}{t}=\frac{ES}{C}
  8. E E
  9. C C
  10. K K
  11. C C
  12. K K
  13. K K
  14. K K
  15. C C
  16. C C
  17. C C
  18. C C
  19. L E = K C \frac{L}{E}=\frac{K}{C}
  20. R R
  21. R = flux emitted from surface flux incident upon surface R=\frac{\mbox{flux emitted from surface}}{\mbox{flux incident upon surface}}
  22. L L
  23. π \pi
  24. L L
  25. R = π L E = π K C R=\frac{\pi L}{E}=\frac{\pi K}{C}
  26. K K
  27. C C
  28. R = π × 12.5 250 15.7 % R=\frac{\pi\times 12.5}{250}\approx 15.7\%
  29. K K
  30. K K
  31. C C
  32. R = π × 12.5 330 11.9 % R=\frac{\pi\times 12.5}{330}\approx 11.9\%
  33. average scene reflectance = average scene luminance effective scene illuminance \mbox{average scene reflectance}~{}=\frac{\mbox{average scene luminance}}{% \mbox{effective scene illuminance }}
  34. K K
  35. K K

Light_value.html

  1. B v B_{v}
  2. L v L_{v}
  3. B v B_{v}

Limit_of_a_function.html

  1. sin x x \frac{\sin x}{x}
  2. y = f ( x ) y=f(x)
  3. lim x p f ( x ) = L , \lim_{x\to p}f(x)=L,\,
  4. lim x p f ( x ) = L , \lim_{x\to p}f(x)=L,
  5. α \alpha
  6. lim x p + f ( x ) = L \lim_{x\to p^{+}}f(x)=L
  7. lim x p - f ( x ) = L \lim_{x\to p^{-}}f(x)=L
  8. ( - , a ) (-\infty,a)
  9. ( a , ) (a,\infty)
  10. f ( x ) = { sin 5 x - 1 for x < 1 0 for x = 1 0.1 x - 1 for x > 1 f(x)=\begin{cases}\sin\frac{5}{x-1}&\,\text{ for }x<1\\ 0&\,\text{ for }x=1\\ \frac{0.1}{x-1}&\,\text{ for }x>1\end{cases}
  11. x 0 = 1 x_{0}=1
  12. f ( x ) = { 1 x rational 0 x irrational f(x)=\begin{cases}1&x\,\text{ rational }\\ 0&x\,\text{ irrational }\end{cases}
  13. f ( x ) = { 1 for x < 0 2 for x 0 f(x)=\begin{cases}1&\,\text{ for }x<0\\ 2&\,\text{ for }x\geq 0\end{cases}
  14. f ( x ) = { x x rational 0 x irrational f(x)=\begin{cases}x&x\,\text{ rational }\\ 0&x\,\text{ irrational }\end{cases}
  15. f ( x ) = { | x | x rational 0 x irrational f(x)=\begin{cases}|x|&x\,\text{ rational }\\ 0&x\,\text{ irrational }\end{cases}
  16. f ( x ) = { sin x x irrational 1 x rational f(x)=\begin{cases}\sin x&x\,\text{ irrational }\\ 1&x\,\text{ rational }\end{cases}
  17. π 2 + 2 n π \frac{\pi}{2}+2n\pi
  18. lim x p f ( x ) = L \lim_{x\to p}f(x)=L
  19. lim x p f ( x ) = L \lim_{x\to p}f(x)=L
  20. lim x p f ( x ) = L \lim_{x\to p}f(x)=L
  21. lim x f ( x ) = L , \lim_{x\to\infty}f(x)=L,
  22. ε > 0 \varepsilon>0
  23. | f ( x ) - L | < ε |f(x)-L|<\varepsilon
  24. ε > 0 c x > c : | f ( x ) - L | < ε \forall\varepsilon>0\;\exists c\;\forall x>c:\;|f(x)-L|<\varepsilon
  25. lim x - f ( x ) = L , \lim_{x\to-\infty}f(x)=L,
  26. ε > 0 \varepsilon>0
  27. | f ( x ) - L | < ε |f(x)-L|<\varepsilon
  28. lim x - e x = 0. \lim_{x\to-\infty}e^{x}=0.\,
  29. lim x a f ( x ) = , \lim_{x\to a}f(x)=\infty,\,
  30. ε > 0 \varepsilon>0
  31. δ > 0 \delta>0
  32. f ( x ) > ε f(x)>\varepsilon
  33. | x - a | < δ |x-a|<\delta
  34. lim x f ( x ) = , lim x a + f ( x ) = - . \lim_{x\to\infty}f(x)=\infty,\lim_{x\to a^{+}}f(x)=-\infty.\,
  35. lim x 0 + ln x = - . \lim_{x\to 0^{+}}\ln x=-\infty.\,
  36. x - 1 x^{-1}
  37. lim x 0 + 1 x = + , lim x 0 - 1 x = - . \lim_{x\to 0^{+}}{1\over x}=+\infty,\lim_{x\to 0^{-}}{1\over x}=-\infty.
  38. lim x 0 + 1 x = lim x 0 - 1 x = lim x 0 1 x = . \lim_{x\to 0^{+}}{1\over x}=\lim_{x\to 0^{-}}{1\over x}=\lim_{x\to 0}{1\over x% }=\infty.
  39. lim x 0 - x - 1 = - \lim_{x\to 0^{-}}{x^{-1}}=-\infty
  40. lim x - x - 1 = - 0 \lim_{x\to-\infty}{x^{-1}}=-0
  41. lim ( x , y ) ( p , q ) f ( x , y ) = L \lim_{(x,y)\to(p,q)}f(x,y)=L
  42. lim x a f ( x ) = L \lim_{x\to a}f(x)=L
  43. x n x_{n}
  44. x n x_{n}
  45. a a
  46. f ( x n ) f(x_{n})
  47. L L
  48. x n x_{n}
  49. a a
  50. x n x_{n}
  51. x n x_{n}
  52. f ( x n ) f(x_{n})
  53. L L
  54. lim x a f ( x ) = L \lim_{x\to a}f(x)=L
  55. x * x\in\mathbb{R}^{*}
  56. f * ( x ) - L f^{*}(x)-L
  57. x - a x-a
  58. * \mathbb{R}^{*}
  59. f * f^{*}
  60. x x
  61. A A\subseteq\mathbb{R}
  62. r > 0 r>0
  63. a A a\in A
  64. | x - a | < r |x-a|<r
  65. lim x a f ( x ) = L \lim_{x\to a}f(x)=L
  66. A A\subseteq\mathbb{R}
  67. L L
  68. f ( A ) f(A)
  69. a a
  70. A A
  71. f ( A ) f(A)
  72. { y y = f ( x ) } \{y\in\mathbb{R}\mid y=f(x)\}
  73. lim x c f ( x ) = f ( c ) . \lim_{x\to c}f(x)=f(c).
  74. lim y b f ( y ) = c \lim_{y\to b}f(y)=c
  75. lim x a g ( x ) = b lim x a f ( g ( x ) ) = c \lim_{x\to a}g(x)=b\Rightarrow\lim_{x\to a}f(g(x))=c
  76. δ > 0 \delta>0
  77. 0 < | x - a | < δ 0<|x-a|<\delta
  78. | g ( x ) - b | > 0 |g(x)-b|>0
  79. f ( x ) = g ( x ) = { 0 if x 0 1 if x = 0 f(x)=g(x)=\begin{cases}0&\,\text{if }x\neq 0\\ 1&\,\text{if }x=0\end{cases}
  80. lim x a f ( x ) = 0 \lim_{x\to a}f(x)=0
  81. a a
  82. f ( f ( x ) ) = { 1 if x 0 0 if x = 0 f(f(x))=\begin{cases}1&\,\text{if }x\neq 0\\ 0&\,\text{if }x=0\end{cases}
  83. lim x a f ( f ( x ) ) = 1 \lim_{x\to a}f(f(x))=1
  84. a a
  85. lim x 0 sin x x = 1 \lim_{x\to 0}\frac{\sin x}{x}=1
  86. lim x 0 1 - cos x x = 0 \lim_{x\to 0}\frac{1-\cos x}{x}=0
  87. lim x x sin ( c x ) = c \lim_{x\to\infty}x\sin\left(\frac{c}{x}\right)=c
  88. 1 < x sin x < tan x sin x 1<\frac{x}{\sin x}<\frac{\tan x}{\sin x}
  89. 1 < x sin x < 1 cos x 1<\frac{x}{\sin x}<\frac{1}{\cos x}
  90. lim x 0 1 cos x = 1 1 = 1 \lim_{x\to 0}\frac{1}{\cos x}=\frac{1}{1}=1
  91. lim x 0 x sin x = 1 \lim_{x\to 0}\frac{x}{\sin x}=1
  92. lim x 0 sin x x = 1 \lim_{x\to 0}\frac{\sin x}{x}=1
  93. 1 - cos 2 x = sin 2 x 1-\cos^{2}x=\sin^{2}x
  94. 1 - cos x x \frac{1-\cos x}{x}
  95. ( 1 - cos x ) ( 1 + cos x ) x ( 1 + cos x ) = ( 1 - cos 2 x ) x ( 1 + cos x ) = sin 2 x x ( 1 + cos x ) = sin x x sin x 1 + cos x \frac{(1-\cos x)(1+\cos x)}{x(1+\cos x)}=\frac{(1-\cos^{2}x)}{x(1+\cos x)}=% \frac{\sin^{2}x}{x(1+\cos x)}=\frac{\sin x}{x}\frac{\sin x}{1+\cos x}
  96. lim x 0 ( sin x x sin x 1 + cos x ) = ( lim x 0 sin x x ) ( lim x 0 sin x 1 + cos x ) = ( 1 ) ( 0 2 ) = 0 \lim_{x\to 0}\left(\frac{\sin x}{x}\frac{\sin x}{1+\cos x}\right)=\left(\lim_{% x\to 0}\frac{\sin x}{x}\right)\left(\lim_{x\to 0}\frac{\sin x}{1+\cos x}\right% )=\left(1\right)\left(\frac{0}{2}\right)=0
  97. lim x 0 1 - cos x x = 0 \lim_{x\to 0}\frac{1-\cos x}{x}=0
  98. 0 / 0 0/0
  99. ± / ±∞/∞
  100. f ( x ) f(x)
  101. g ( x ) g(x)
  102. I I
  103. lim x c f ( x ) = lim x c g ( x ) = 0 , or lim x c f ( x ) = ± lim x c g ( x ) = ± , and \lim_{x\to c}f(x)=\lim_{x\to c}g(x)=0,\,\text{ or }\lim_{x\to c}f(x)=\pm\lim_{% x\to c}g(x)=\pm\infty,\,\text{and}
  104. f and g are differentiable over I { c } , and f\,\text{ and }g\,\text{ are differentiable over }I\setminus\{c\},\,\text{ and}
  105. g ( x ) 0 for all x I { c } , and g^{\prime}(x)\neq 0\,\text{ for all }x\in I\setminus\{c\},\,\text{ and}
  106. lim x c f ( x ) g ( x ) exists, \lim_{x\to c}\frac{f^{\prime}(x)}{g^{\prime}(x)}\,\text{ exists,}
  107. lim x c f ( x ) g ( x ) = lim x c f ( x ) g ( x ) \lim_{x\to c}\frac{f(x)}{g(x)}=\lim_{x\to c}\frac{f^{\prime}(x)}{g^{\prime}(x)}
  108. lim x 0 sin ( 2 x ) sin ( 3 x ) = lim x 0 2 cos ( 2 x ) 3 cos ( 3 x ) = 2 \sdot 1 3 \sdot 1 = 2 3 . \lim_{x\to 0}\frac{\sin(2x)}{\sin(3x)}=\lim_{x\to 0}\frac{2\cos(2x)}{3\cos(3x)% }=\frac{2\sdot 1}{3\sdot 1}=\frac{2}{3}.
  109. lim n i = s n f ( i ) \lim_{n\to\infty}\sum_{i=s}^{n}f(i)
  110. i = s f ( i ) \sum_{i=s}^{\infty}f(i)
  111. lim x a x f ( t ) d t \lim_{x\to\infty}\int_{a}^{x}f(t)\;dt
  112. a f ( t ) d t \int_{a}^{\infty}f(t)\;dt
  113. lim x - x b f ( t ) d t \lim_{x\to-\infty}\int_{x}^{b}f(t)\;dt
  114. - b f ( t ) d t \int_{-\infty}^{b}f(t)\;dt

Limit_of_a_sequence.html

  1. 2 π r 2\pi r
  2. L L
  3. ( x n ) (x_{n})
  4. L L
  5. x n = c x_{n}=c
  6. x n c x_{n}\to c
  7. x n = 1 / n x_{n}=1/n
  8. x n 0 x_{n}\to 0
  9. x n = 1 / n x_{n}=1/n
  10. n n
  11. x n = 1 / n 2 x_{n}=1/n^{2}
  12. n n
  13. x n 0 x_{n}\to 0
  14. x n + 1 > x n x_{n+1}>x_{n}
  15. n n
  16. 0.3 , 0.33 , 0.333 , 0.3333 , 0.3,0.33,0.333,0.3333,...
  17. 1 / 3 1/3
  18. 0.3333... 0.3333...
  19. 0.3333... lim n i = 1 n 3 10 i 0.3333...\triangleq\lim_{n\to\infty}\sum_{i=1}^{n}\frac{3}{10^{i}}
  20. lim n ( 1 + 1 n ) n \lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n}
  21. x x
  22. ( x n ) (x_{n})
  23. ϵ > 0 \epsilon>0
  24. N N
  25. n > N n>N
  26. | x n - x | < ϵ |x_{n}-x|<\epsilon
  27. ϵ \epsilon
  28. ( x n ) (x_{n})
  29. x x
  30. x n x x_{n}\to x
  31. lim n x n = x \lim_{n\to\infty}x_{n}=x
  32. a n a a_{n}\to a
  33. b n b b_{n}\to b
  34. a n + b n a + b a_{n}+b_{n}\to a+b
  35. a n b n a b a_{n}b_{n}\to ab
  36. b n b_{n}
  37. a n / b n a / b a_{n}/b_{n}\to a/b
  38. x n x x_{n}\to x
  39. f ( x n ) f ( x ) f(x_{n})\to f(x)
  40. lim n ( a n ± b n ) = lim n a n ± lim n b n \lim_{n\to\infty}(a_{n}\pm b_{n})=\lim_{n\to\infty}a_{n}\pm\lim_{n\to\infty}b_% {n}
  41. lim n c a n = c lim n a n \lim_{n\to\infty}ca_{n}=c\lim_{n\to\infty}a_{n}
  42. lim n ( a n b n ) = ( lim n a n ) ( lim n b n ) \lim_{n\to\infty}(a_{n}b_{n})=(\lim_{n\to\infty}a_{n})(\lim_{n\to\infty}b_{n})
  43. lim n a n b n = lim n a n lim n b n \lim_{n\to\infty}\frac{a_{n}}{b_{n}}=\frac{\lim_{n\to\infty}a_{n}}{\lim_{n\to% \infty}b_{n}}
  44. lim n b n 0 \lim_{n\to\infty}b_{n}\neq 0
  45. lim n a n p = [ lim n a n ] p \lim_{n\to\infty}a_{n}^{p}=\left[\lim_{n\to\infty}a_{n}\right]^{p}
  46. a n b n a_{n}\leq b_{n}
  47. n n
  48. N N
  49. lim n a n lim n b n \lim_{n\to\infty}a_{n}\leq\lim_{n\to\infty}b_{n}
  50. a n c n b n a_{n}\leq c_{n}\leq b_{n}
  51. n > N n>N
  52. lim n a n = lim n b n = L \lim_{n\to\infty}a_{n}=\lim_{n\to\infty}b_{n}=L
  53. lim n c n = L \lim_{n\to\infty}c_{n}=L
  54. 1 / n 0 1/n\to 0
  55. a b + c / n a b \frac{a}{b+c/n}\to\frac{a}{b}
  56. b 0 b\neq 0
  57. ( x n ) (x_{n})
  58. x n x_{n}\to\infty
  59. lim n x n = \lim_{n\to\infty}x_{n}=\infty
  60. n N n\geq N
  61. x n > K x_{n}>K
  62. x n - x_{n}\to-\infty
  63. n N n\geq N
  64. x n < K x_{n}<K
  65. n N n\geq N
  66. d ( x n , x ) < ϵ d(x_{n},x)<\epsilon
  67. X = X=\mathbb{R}
  68. d ( x , y ) = | x - y | d(x,y)=|x-y|
  69. x n x x_{n}\to x
  70. f ( x n ) f ( x ) f(x_{n})\to f(x)
  71. ϵ \epsilon
  72. ϵ \epsilon
  73. n N n\geq N
  74. x n U x_{n}\in U
  75. τ \tau
  76. ( x n : n ) \left(x_{n}:n\in\mathbb{N}\right)\;
  77. \mathbb{N}
  78. { + } \mathbb{N}\cup\{+\infty\}
  79. \mathbb{N}
  80. ( x n ) (x_{n})
  81. L = st ( x H ) L={\rm st}(x_{H})\,
  82. lim n x n = st ( x H ) , \lim_{n\to\infty}x_{n}={\rm st}(x_{H}),
  83. N = 1 N=1
  84. n > N n>N
  85. | x n - c | = 0 < ϵ |x_{n}-c|=0<\epsilon
  86. N = 1 ϵ N=\left\lfloor\frac{1}{\epsilon}\right\rfloor
  87. n > N n>N
  88. | x n - 0 | x N + 1 = 1 1 / ϵ + 1 < ϵ |x_{n}-0|\leq x_{N+1}=\frac{1}{\lfloor 1/\epsilon\rfloor+1}<\epsilon

Limit_point.html

  1. n n + 1 n\frac{n}{+1}
  2. n : ( P , ) X n:(P,\leq)\to X
  3. ( P , ) (P,\leq)
  4. a X a\in X
  5. n n
  6. U U
  7. a a
  8. p P p\in P
  9. x p x\geq p
  10. n ( x ) U n(x)\in U
  11. n n
  12. a a

Linati_schema_for_Ulysses.html

  1. \infty

Lindelöf_space.html

  1. 𝕊 \mathbb{S}
  2. \mathbb{R}
  3. 𝕊 \mathbb{S}
  4. ( x , y ) (x,y)
  5. x + y = 0 x+y=0
  6. 𝕊 \mathbb{S}
  7. ( - , x ) × ( - , y ) (-\infty,x)\times(-\infty,y)
  8. ( x , y ) (x,y)
  9. [ x , + ) × [ y , + ) [x,+\infty)\times[y,+\infty)
  10. ( x , y ) (x,y)
  11. S S
  12. S S
  13. κ \kappa
  14. κ \kappa
  15. κ \kappa
  16. κ \kappa
  17. 0 \aleph_{0}
  18. 1 \aleph_{1}
  19. l ( X ) l(X)
  20. κ \kappa
  21. X X
  22. κ \kappa
  23. X X
  24. l ( X ) = 0 l(X)=\aleph_{0}
  25. κ \kappa
  26. X X
  27. κ \kappa
  28. κ \kappa
  29. X X
  30. κ \kappa
  31. X X

Line_drawing_algorithm.html

  1. Y = m x + b Y=mx+b
  2. ( x 1 , y 1 ) (x1,y1)
  3. ( x 2 , y 2 ) (x2,y2)
  4. m = ( y 2 - y 1 ) / ( x 2 - x 1 ) m=(y2-y1)/(x2-x1)
  5. b = y 1 - m . x 1 b=y1-m.x1
  6. x 2 > x 1 x_{2}>x_{1}
  7. d x d y dx>=dy
  8. d x < d y dx<dy
  9. d x = 0 dx=0

Linear_classifier.html

  1. x \vec{x}
  2. y = f ( w x ) = f ( j w j x j ) , y=f(\vec{w}\cdot\vec{x})=f\left(\sum_{j}w_{j}x_{j}\right),
  3. w \vec{w}
  4. w \vec{w}
  5. x \vec{x}
  6. w \vec{w}
  7. x \vec{x}
  8. x \vec{x}
  9. x \vec{x}
  10. w \vec{w}
  11. P ( x | class ) P(\vec{x}|{\rm class})
  12. w \vec{w}
  13. φ ( x ) \varphi(\vec{x})
  14. arg min 𝐰 R ( 𝐰 ) + C i = 1 N L ( y i , 𝐰 𝖳 𝐱 i ) \arg\min_{\mathbf{w}}R(\mathbf{w})+C\sum_{i=1}^{N}L(y_{i},\mathbf{w}^{\mathsf{% T}}\mathbf{x}_{i})
  15. 𝐰 \mathbf{w}
  16. L ( y < s u b > i , 𝐰 T 𝐱 i ) L(y<sub>i,\mathbf{w}^{T}\mathbf{x}_{i})

Linear_elasticity.html

  1. s y m b o l \cdotsymbol σ + 𝐅 = ρ 𝐮 ¨ symbol{\nabla}\cdotsymbol{\sigma}+\mathbf{F}=\rho\ddot{\mathbf{u}}
  2. s y m b o l ε = 1 2 [ s y m b o l 𝐮 + ( s y m b o l 𝐮 ) T ] symbol{\varepsilon}=\tfrac{1}{2}\left[symbol{\nabla}\mathbf{u}+(symbol{\nabla}% \mathbf{u})^{T}\right]\,\!
  3. s y m b o l σ = 𝖢 : s y m b o l ε , symbol{\sigma}=\mathsf{C}:symbol{\varepsilon},
  4. s y m b o l σ symbol{\sigma}
  5. s y m b o l ε symbol{\varepsilon}
  6. 𝐮 \mathbf{u}
  7. 𝖢 \mathsf{C}
  8. 𝐅 \mathbf{F}
  9. ρ \rho
  10. s y m b o l symbol{\nabla}
  11. ( ) T (\bullet)^{T}
  12. ( ) ¨ \ddot{(\bullet)}
  13. 𝐀 : 𝐁 = A i j B i j \mathbf{A}:\mathbf{B}=A_{ij}B_{ij}
  14. σ j i , j + F i = ρ t t u i \sigma_{ji,j}+F_{i}=\rho\partial_{tt}u_{i}\,\!
  15. σ x x + τ y x y + τ z x z + F x = ρ 2 u x t 2 \frac{\partial\sigma_{x}}{\partial x}+\frac{\partial\tau_{yx}}{\partial y}+% \frac{\partial\tau_{zx}}{\partial z}+F_{x}=\rho\frac{\partial^{2}u_{x}}{% \partial t^{2}}\,\!
  16. τ x y x + σ y y + τ z y z + F y = ρ 2 u y t 2 \frac{\partial\tau_{xy}}{\partial x}+\frac{\partial\sigma_{y}}{\partial y}+% \frac{\partial\tau_{zy}}{\partial z}+F_{y}=\rho\frac{\partial^{2}u_{y}}{% \partial t^{2}}\,\!
  17. τ x z x + τ y z y + σ z z + F z = ρ 2 u z t 2 \frac{\partial\tau_{xz}}{\partial x}+\frac{\partial\tau_{yz}}{\partial y}+% \frac{\partial\sigma_{z}}{\partial z}+F_{z}=\rho\frac{\partial^{2}u_{z}}{% \partial t^{2}}\,\!
  18. ( ) , j {(\bullet)}_{,j}
  19. ( ) / x j \partial{(\bullet)}/\partial x_{j}
  20. t t \partial_{tt}
  21. 2 / t 2 \partial^{2}/\partial t^{2}
  22. σ i j = σ j i \sigma_{ij}=\sigma_{ji}\,\!
  23. F i F_{i}\,\!
  24. ρ \rho\,\!
  25. u i u_{i}\,\!
  26. ε i j = 1 2 ( u j , i + u i , j ) \varepsilon_{ij}=\frac{1}{2}(u_{j,i}+u_{i,j})\,\!
  27. ϵ x = u x x \epsilon_{x}=\frac{\partial u_{x}}{\partial x}\,\!
  28. γ x y = u x y + u y x \gamma_{xy}=\frac{\partial u_{x}}{\partial y}+\frac{\partial u_{y}}{\partial x% }\,\!
  29. ϵ y = u y y \epsilon_{y}=\frac{\partial u_{y}}{\partial y}\,\!
  30. γ y z = u y z + u z y \gamma_{yz}=\frac{\partial u_{y}}{\partial z}+\frac{\partial u_{z}}{\partial y% }\,\!
  31. ϵ z = u z z \epsilon_{z}=\frac{\partial u_{z}}{\partial z}\,\!
  32. γ z x = u z x + u x z \gamma_{zx}=\frac{\partial u_{z}}{\partial x}+\frac{\partial u_{x}}{\partial z% }\,\!
  33. ε i j = ε j i \varepsilon_{ij}=\varepsilon_{ji}\,\!
  34. σ i j = C i j k l ε k l \sigma_{ij}=C_{ijkl}\,\varepsilon_{kl}\,\!
  35. C i j k l C_{ijkl}
  36. C i j k l = C k l i j = C j i k l = C i j l k C_{ijkl}=C_{klij}=C_{jikl}=C_{ijlk}
  37. r , θ , z r,\theta,z
  38. σ r r r + 1 r σ r θ θ + σ r z z + 1 r ( σ r r - σ θ θ ) + F r = ρ 2 u r t 2 σ r θ r + 1 r σ θ θ θ + σ θ z z + 2 r σ r θ + F θ = ρ 2 u θ t 2 σ r z r + 1 r σ θ z θ + σ z z z + 1 r σ r z + F z = ρ 2 u z t 2 \begin{aligned}&\displaystyle\frac{\partial\sigma_{rr}}{\partial r}+\cfrac{1}{% r}\frac{\partial\sigma_{r\theta}}{\partial\theta}+\frac{\partial\sigma_{rz}}{% \partial z}+\cfrac{1}{r}(\sigma_{rr}-\sigma_{\theta\theta})+F_{r}=\rho~{}\frac% {\partial^{2}u_{r}}{\partial t^{2}}\\ &\displaystyle\frac{\partial\sigma_{r\theta}}{\partial r}+\cfrac{1}{r}\frac{% \partial\sigma_{\theta\theta}}{\partial\theta}+\frac{\partial\sigma_{\theta z}% }{\partial z}+\cfrac{2}{r}\sigma_{r\theta}+F_{\theta}=\rho~{}\frac{\partial^{2% }u_{\theta}}{\partial t^{2}}\\ &\displaystyle\frac{\partial\sigma_{rz}}{\partial r}+\cfrac{1}{r}\frac{% \partial\sigma_{\theta z}}{\partial\theta}+\frac{\partial\sigma_{zz}}{\partial z% }+\cfrac{1}{r}\sigma_{rz}+F_{z}=\rho~{}\frac{\partial^{2}u_{z}}{\partial t^{2}% }\end{aligned}
  39. ε r r = u r r ; ε θ θ = 1 r ( u θ θ + u r ) ; ε z z = u z z ε r θ = 1 2 ( 1 r u r θ + u θ r - u θ r ) ; ε θ z = 1 2 ( u θ z + 1 r u z θ ) ; ε z r = 1 2 ( u r z + u z r ) \begin{aligned}\displaystyle\varepsilon_{rr}&\displaystyle=\cfrac{\partial u_{% r}}{\partial r}~{};~{}~{}\varepsilon_{\theta\theta}=\cfrac{1}{r}\left(\cfrac{% \partial u_{\theta}}{\partial\theta}+u_{r}\right)~{};~{}~{}\varepsilon_{zz}=% \cfrac{\partial u_{z}}{\partial z}\\ \displaystyle\varepsilon_{r\theta}&\displaystyle=\cfrac{1}{2}\left(\cfrac{1}{r% }\cfrac{\partial u_{r}}{\partial\theta}+\cfrac{\partial u_{\theta}}{\partial r% }-\cfrac{u_{\theta}}{r}\right)~{};~{}~{}\varepsilon_{\theta z}=\cfrac{1}{2}% \left(\cfrac{\partial u_{\theta}}{\partial z}+\cfrac{1}{r}\cfrac{\partial u_{z% }}{\partial\theta}\right)~{};~{}~{}\varepsilon_{zr}=\cfrac{1}{2}\left(\cfrac{% \partial u_{r}}{\partial z}+\cfrac{\partial u_{z}}{\partial r}\right)\end{aligned}
  40. 1 1
  41. 2 2
  42. 3 3
  43. r r
  44. θ \theta
  45. z z
  46. r , θ , ϕ r,\theta,\phi
  47. σ r r r + 1 r σ r θ θ + 1 r sin θ σ r ϕ ϕ + 1 r ( 2 σ r r - σ θ θ - σ ϕ ϕ + σ r θ cot θ ) + F r = ρ 2 u r t 2 σ r θ r + 1 r σ θ θ θ + 1 r sin θ σ θ ϕ ϕ + 1 r [ ( σ θ θ - σ ϕ ϕ ) cot θ + 3 σ r θ ] + F θ = ρ 2 u θ t 2 σ r ϕ r + 1 r σ θ ϕ θ + 1 r sin θ σ ϕ ϕ ϕ + 1 r ( 2 σ θ ϕ cot θ + 3 σ r ϕ ) + F ϕ = ρ 2 u ϕ t 2 \begin{aligned}&\displaystyle\frac{\partial\sigma_{rr}}{\partial r}+\cfrac{1}{% r}\frac{\partial\sigma_{r\theta}}{\partial\theta}+\cfrac{1}{r\sin\theta}\frac{% \partial\sigma_{r\phi}}{\partial\phi}+\cfrac{1}{r}(2\sigma_{rr}-\sigma_{\theta% \theta}-\sigma_{\phi\phi}+\sigma_{r\theta}\cot\theta)+F_{r}=\rho~{}\frac{% \partial^{2}u_{r}}{\partial t^{2}}\\ &\displaystyle\frac{\partial\sigma_{r\theta}}{\partial r}+\cfrac{1}{r}\frac{% \partial\sigma_{\theta\theta}}{\partial\theta}+\cfrac{1}{r\sin\theta}\frac{% \partial\sigma_{\theta\phi}}{\partial\phi}+\cfrac{1}{r}[(\sigma_{\theta\theta}% -\sigma_{\phi\phi})\cot\theta+3\sigma_{r\theta}]+F_{\theta}=\rho~{}\frac{% \partial^{2}u_{\theta}}{\partial t^{2}}\\ &\displaystyle\frac{\partial\sigma_{r\phi}}{\partial r}+\cfrac{1}{r}\frac{% \partial\sigma_{\theta\phi}}{\partial\theta}+\cfrac{1}{r\sin\theta}\frac{% \partial\sigma_{\phi\phi}}{\partial\phi}+\cfrac{1}{r}(2\sigma_{\theta\phi}\cot% \theta+3\sigma_{r\phi})+F_{\phi}=\rho~{}\frac{\partial^{2}u_{\phi}}{\partial t% ^{2}}\end{aligned}
  48. ε r r = u r r ε θ θ = 1 r ( u θ θ + u r ) ε ϕ ϕ = 1 r sin θ ( u ϕ ϕ + u r sin θ + u θ cos θ ) ε r θ = 1 2 ( 1 r u r θ + u θ r - u θ r ) ε θ ϕ = 1 2 r [ 1 sin θ u θ ϕ + ( u ϕ θ - u ϕ cot θ ) ] ε r ϕ = 1 2 ( 1 r sin θ u r ϕ + u ϕ r - u ϕ r ) . \begin{aligned}\displaystyle\varepsilon_{rr}&\displaystyle=\frac{\partial u_{r% }}{\partial r}\\ \displaystyle\varepsilon_{\theta\theta}&\displaystyle=\frac{1}{r}\left(\frac{% \partial u_{\theta}}{\partial\theta}+u_{r}\right)\\ \displaystyle\varepsilon_{\phi\phi}&\displaystyle=\frac{1}{r\sin\theta}\left(% \frac{\partial u_{\phi}}{\partial\phi}+u_{r}\sin\theta+u_{\theta}\cos\theta% \right)\\ \displaystyle\varepsilon_{r\theta}&\displaystyle=\frac{1}{2}\left(\frac{1}{r}% \frac{\partial u_{r}}{\partial\theta}+\frac{\partial u_{\theta}}{\partial r}-% \frac{u_{\theta}}{r}\right)\\ \displaystyle\varepsilon_{\theta\phi}&\displaystyle=\frac{1}{2r}\left[\frac{1}% {\sin\theta}\frac{\partial u_{\theta}}{\partial\phi}+\left(\frac{\partial u_{% \phi}}{\partial\theta}-u_{\phi}\cot\theta\right)\right]\\ \displaystyle\varepsilon_{r\phi}&\displaystyle=\frac{1}{2}\left(\frac{1}{r\sin% \theta}\frac{\partial u_{r}}{\partial\phi}+\frac{\partial u_{\phi}}{\partial r% }-\frac{u_{\phi}}{r}\right).\end{aligned}
  49. C i j k l = K δ i j δ k l + μ ( δ i k δ j l + δ i l δ j k - 2 3 δ i j δ k l ) C_{ijkl}=K\,\delta_{ij}\,\delta_{kl}+\mu\,(\delta_{ik}\delta_{jl}+\delta_{il}% \delta_{jk}-\textstyle{\frac{2}{3}}\,\delta_{ij}\,\delta_{kl})\,\!
  50. δ i j \delta_{ij}\,\!
  51. μ \mu\,\!
  52. σ i j = K δ i j ε k k + 2 μ ( ε i j - 1 3 δ i j ε k k ) . \sigma_{ij}=K\delta_{ij}\varepsilon_{kk}+2\mu(\varepsilon_{ij}-\textstyle{% \frac{1}{3}}\delta_{ij}\varepsilon_{kk}).\,\!
  53. σ i j = λ δ i j ε k k + 2 μ ε i j \sigma_{ij}=\lambda\delta_{ij}\varepsilon_{kk}+2\mu\varepsilon_{ij}\,\!
  54. ε i j = 1 9 K δ i j σ k k + 1 2 μ ( σ i j - 1 3 δ i j σ k k ) \varepsilon_{ij}=\frac{1}{9K}\delta_{ij}\sigma_{kk}+\frac{1}{2\mu}\left(\sigma% _{ij}-\textstyle{\frac{1}{3}}\delta_{ij}\sigma_{kk}\right)\,\!
  55. ε i j = 1 2 μ σ i j - ν E δ i j σ k k = 1 E [ ( 1 + ν ) σ i j - ν δ i j σ k k ] \varepsilon_{ij}=\frac{1}{2\mu}\sigma_{ij}-\frac{\nu}{E}\delta_{ij}\sigma_{kk}% =\frac{1}{E}[(1+\nu)\sigma_{ij}-\nu\delta_{ij}\sigma_{kk}]\,\!
  56. σ j i , j + F i = 0. \sigma_{ji,j}+F_{i}=0.\,\!
  57. σ x x + τ y x y + τ z x z + F x = 0 \frac{\partial\sigma_{x}}{\partial x}+\frac{\partial\tau_{yx}}{\partial y}+% \frac{\partial\tau_{zx}}{\partial z}+F_{x}=0\,\!
  58. τ x y x + σ y y + τ z y z + F y = 0 \frac{\partial\tau_{xy}}{\partial x}+\frac{\partial\sigma_{y}}{\partial y}+% \frac{\partial\tau_{zy}}{\partial z}+F_{y}=0\,\!
  59. τ x z x + τ y z y + σ z z + F z = 0 \frac{\partial\tau_{xz}}{\partial x}+\frac{\partial\tau_{yz}}{\partial y}+% \frac{\partial\sigma_{z}}{\partial z}+F_{z}=0\,\!
  60. σ i j = λ δ i j ε k k + 2 μ ε i j = λ δ i j u k , k + μ ( u i , j + u j , i ) . \begin{aligned}\displaystyle\sigma_{ij}&\displaystyle=\lambda\delta_{ij}% \varepsilon_{kk}+2\mu\varepsilon_{ij}\\ &\displaystyle=\lambda\delta_{ij}u_{k,k}+\mu\left(u_{i,j}+u_{j,i}\right).\\ \end{aligned}\,\!
  61. σ i j , j = λ u k , k i + μ ( u i , j j + u j , i j ) . \sigma_{ij,j}=\lambda u_{k,ki}+\mu\left(u_{i,jj}+u_{j,ij}\right).\,\!
  62. λ u k , k i + μ ( u i , j j + u j , i j ) + F i = 0 \lambda u_{k,ki}+\mu\left(u_{i,jj}+u_{j,ij}\right)+F_{i}=0\,\!
  63. μ u i , j j + ( μ + λ ) u j , i j + F i = 0 \mu u_{i,jj}+(\mu+\lambda)u_{j,ij}+F_{i}=0\,\!
  64. λ \lambda\,\!
  65. μ \mu\,\!
  66. x x\,\!
  67. x x\,\!
  68. σ x = 2 μ ε x + λ ( ε x + ε y + ε z ) = 2 μ u x x + λ ( u x x + u y y + u z z ) \sigma_{x}=2\mu\varepsilon_{x}+\lambda(\varepsilon_{x}+\varepsilon_{y}+% \varepsilon_{z})=2\mu\frac{\partial u_{x}}{\partial x}+\lambda\left(\frac{% \partial u_{x}}{\partial x}+\frac{\partial u_{y}}{\partial y}+\frac{\partial u% _{z}}{\partial z}\right)\,\!
  69. τ x y = μ γ x y = μ ( u x y + u y x ) \tau_{xy}=\mu\gamma_{xy}=\mu\left(\frac{\partial u_{x}}{\partial y}+\frac{% \partial u_{y}}{\partial x}\right)\,\!
  70. τ x z = μ γ z x = μ ( u z x + u x z ) \tau_{xz}=\mu\gamma_{zx}=\mu\left(\frac{\partial u_{z}}{\partial x}+\frac{% \partial u_{x}}{\partial z}\right)\,\!
  71. x x\,\!
  72. σ x x + τ y x y + τ z x z + F x = 0 \frac{\partial\sigma_{x}}{\partial x}+\frac{\partial\tau_{yx}}{\partial y}+% \frac{\partial\tau_{zx}}{\partial z}+F_{x}=0\,\!
  73. x ( 2 μ u x x + λ ( u x x + u y y + u z z ) ) + μ y ( u x y + u y x ) + μ z ( u z x + u x z ) + F x = 0 \frac{\partial}{\partial x}\left(2\mu\frac{\partial u_{x}}{\partial x}+\lambda% \left(\frac{\partial u_{x}}{\partial x}+\frac{\partial u_{y}}{\partial y}+% \frac{\partial u_{z}}{\partial z}\right)\right)+\mu\frac{\partial}{\partial y}% \left(\frac{\partial u_{x}}{\partial y}+\frac{\partial u_{y}}{\partial x}% \right)+\mu\frac{\partial}{\partial z}\left(\frac{\partial u_{z}}{\partial x}+% \frac{\partial u_{x}}{\partial z}\right)+F_{x}=0\,\!
  74. μ \mu
  75. λ \lambda
  76. ( λ + μ ) x ( u x x + u y y + u z z ) + μ ( 2 u x x 2 + 2 u x y 2 + 2 u x z 2 ) + F x = 0 \left(\lambda+\mu\right)\frac{\partial}{\partial x}\left(\frac{\partial u_{x}}% {\partial x}+\frac{\partial u_{y}}{\partial y}+\frac{\partial u_{z}}{\partial z% }\right)+\mu\left(\frac{\partial^{2}u_{x}}{\partial x^{2}}+\frac{\partial^{2}u% _{x}}{\partial y^{2}}+\frac{\partial^{2}u_{x}}{\partial z^{2}}\right)+F_{x}=0\,\!
  77. y y\,\!
  78. z z\,\!
  79. ( λ + μ ) y ( u x x + u y y + u z z ) + μ ( 2 u y x 2 + 2 u y y 2 + 2 u y z 2 ) + F y = 0 \left(\lambda+\mu\right)\frac{\partial}{\partial y}\left(\frac{\partial u_{x}}% {\partial x}+\frac{\partial u_{y}}{\partial y}+\frac{\partial u_{z}}{\partial z% }\right)+\mu\left(\frac{\partial^{2}u_{y}}{\partial x^{2}}+\frac{\partial^{2}u% _{y}}{\partial y^{2}}+\frac{\partial^{2}u_{y}}{\partial z^{2}}\right)+F_{y}=0\,\!
  80. ( λ + μ ) z ( u x x + u y y + u z z ) + μ ( 2 u z x 2 + 2 u z y 2 + 2 u z z 2 ) + F z = 0 \left(\lambda+\mu\right)\frac{\partial}{\partial z}\left(\frac{\partial u_{x}}% {\partial x}+\frac{\partial u_{y}}{\partial y}+\frac{\partial u_{z}}{\partial z% }\right)+\mu\left(\frac{\partial^{2}u_{z}}{\partial x^{2}}+\frac{\partial^{2}u% _{z}}{\partial y^{2}}+\frac{\partial^{2}u_{z}}{\partial z^{2}}\right)+F_{z}=0\,\!
  81. ( λ + μ ) ( 𝐮 ) + μ 2 𝐮 + 𝐅 = 0 (\lambda+\mu)\nabla(\nabla\cdot\mathbf{u})+\mu\nabla^{2}\mathbf{u}+\mathbf{F}=% 0\,\!
  82. ( α 2 - β 2 ) u j , i j + β 2 u i , m m = - F i . (\alpha^{2}-\beta^{2})u_{j,ij}+\beta^{2}u_{i,mm}=-F_{i}.\,\!
  83. F i , i = 0 F_{i,i}=0\,\!
  84. ( α 2 - β 2 ) u j , i i j + β 2 u i , i m m = 0. (\alpha^{2}-\beta^{2})u_{j,iij}+\beta^{2}u_{i,imm}=0.\,\!
  85. α 2 u j , i i j = 0 \alpha^{2}u_{j,iij}=0\,\!
  86. u j , i i j = 0. u_{j,iij}=0.\,\!
  87. F i , k k = 0 F_{i,kk}=0\,\!
  88. ( α 2 - β 2 ) u j , k k i j + β 2 u i , k k m m = 0. (\alpha^{2}-\beta^{2})u_{j,kkij}+\beta^{2}u_{i,kkmm}=0.\,\!
  89. β 2 u i , k k m m = 0 \beta^{2}u_{i,kkmm}=0\,\!
  90. u i , k k m m = 0 u_{i,kkmm}=0\,\!
  91. 4 𝐮 = 0 \nabla^{4}\mathbf{u}=0\,\!
  92. 𝐮 \mathbf{u}\,\!
  93. ε i j , k m + ε k m , i j - ε i k , j m - ε j m , i k = 0. \varepsilon_{ij,km}+\varepsilon_{km,ij}-\varepsilon_{ik,jm}-\varepsilon_{jm,ik% }=0.\,\!
  94. 2 ϵ x y 2 + 2 ϵ y x 2 = 2 2 ϵ x y x y \frac{\partial^{2}\epsilon_{x}}{\partial y^{2}}+\frac{\partial^{2}\epsilon_{y}% }{\partial x^{2}}=2\frac{\partial^{2}\epsilon_{xy}}{\partial x\partial y}\,\!
  95. 2 ϵ y z 2 + 2 ϵ z y 2 = 2 2 ϵ y z y z \frac{\partial^{2}\epsilon_{y}}{\partial z^{2}}+\frac{\partial^{2}\epsilon_{z}% }{\partial y^{2}}=2\frac{\partial^{2}\epsilon_{yz}}{\partial y\partial z}\,\!
  96. 2 ϵ x z 2 + 2 ϵ z x 2 = 2 2 ϵ z x z x \frac{\partial^{2}\epsilon_{x}}{\partial z^{2}}+\frac{\partial^{2}\epsilon_{z}% }{\partial x^{2}}=2\frac{\partial^{2}\epsilon_{zx}}{\partial z\partial x}\,\!
  97. 2 ϵ x y z = x ( - ϵ y z x + ϵ z x y + ϵ x y z ) \frac{\partial^{2}\epsilon_{x}}{\partial y\partial z}=\frac{\partial}{\partial x% }\left(-\frac{\partial\epsilon_{yz}}{\partial x}+\frac{\partial\epsilon_{zx}}{% \partial y}+\frac{\partial\epsilon_{xy}}{\partial z}\right)\,\!
  98. 2 ϵ y z x = y ( ϵ y z x - ϵ z x y + ϵ x y z ) \frac{\partial^{2}\epsilon_{y}}{\partial z\partial x}=\frac{\partial}{\partial y% }\left(\frac{\partial\epsilon_{yz}}{\partial x}-\frac{\partial\epsilon_{zx}}{% \partial y}+\frac{\partial\epsilon_{xy}}{\partial z}\right)\,\!
  99. 2 ϵ z x y = z ( ϵ y z x + ϵ z x y - ϵ x y z ) \frac{\partial^{2}\epsilon_{z}}{\partial x\partial y}=\frac{\partial}{\partial z% }\left(\frac{\partial\epsilon_{yz}}{\partial x}+\frac{\partial\epsilon_{zx}}{% \partial y}-\frac{\partial\epsilon_{xy}}{\partial z}\right)\,\!
  100. σ i j , k k + 1 1 + ν σ k k , i j + F i , j + F j , i + ν 1 - ν δ i , j F k , k = 0. \sigma_{ij,kk}+\frac{1}{1+\nu}\sigma_{kk,ij}+F_{i,j}+F_{j,i}+\frac{\nu}{1-\nu}% \delta_{i,j}F_{k,k}=0.\,\!
  101. ( 1 + ν ) σ i j , k k + σ k k , i j = 0. (1+\nu)\sigma_{ij,kk}+\sigma_{kk,ij}=0.\,\!
  102. s y m b o l 4 s y m b o l σ = s y m b o l 0 symbol{\nabla}^{4}symbol{\sigma}=symbol{0}
  103. σ i j , k k = 0 \sigma_{ij,kk\ell\ell}=0
  104. a = 1 - 2 ν a=1-2\nu\,\!
  105. b = 2 ( 1 - ν ) = a + 1 b=2(1-\nu)=a+1\,\!
  106. ν \nu\,\!
  107. u i = G i k F k u_{i}=G_{ik}F_{k}\,\!
  108. F k F_{k}\,\!
  109. G i k G_{ik}\,\!
  110. G i k = 1 4 π μ r [ ( 1 - 1 2 b ) δ i k + 1 2 b x i x k r 2 ] G_{ik}=\frac{1}{4\pi\mu r}\left[\left(1-\frac{1}{2b}\right)\delta_{ik}+\frac{1% }{2b}\frac{x_{i}x_{k}}{r^{2}}\right]\,\!
  111. G i k = 1 4 π μ [ δ i k r - 1 2 b 2 r x i x k ] G_{ik}=\frac{1}{4\pi\mu}\left[\frac{\delta_{ik}}{r}-\frac{1}{2b}\frac{\partial% ^{2}r}{\partial x_{i}\partial x_{k}}\right]\,\!
  112. G i k = 1 4 π μ r [ 1 - 1 2 b + 1 2 b x 2 r 2 1 2 b x y r 2 1 2 b x z r 2 1 2 b y x r 2 1 - 1 2 b + 1 2 b y 2 r 2 1 2 b y z r 2 1 2 b z x r 2 1 2 b z y r 2 1 - 1 2 b + 1 2 b z 2 r 2 ] G_{ik}=\frac{1}{4\pi\mu r}\begin{bmatrix}1-\frac{1}{2b}+\frac{1}{2b}\frac{x^{2% }}{r^{2}}&\frac{1}{2b}\frac{xy}{r^{2}}&\frac{1}{2b}\frac{xz}{r^{2}}\\ \frac{1}{2b}\frac{yx}{r^{2}}&1-\frac{1}{2b}+\frac{1}{2b}\frac{y^{2}}{r^{2}}&% \frac{1}{2b}\frac{yz}{r^{2}}\\ \frac{1}{2b}\frac{zx}{r^{2}}&\frac{1}{2b}\frac{zy}{r^{2}}&1-\frac{1}{2b}+\frac% {1}{2b}\frac{z^{2}}{r^{2}}\end{bmatrix}\,\!
  113. ρ , ϕ , z \rho,\phi,z\,\!
  114. G i k = 1 4 π μ r [ 1 - 1 2 b z 2 r 2 0 1 2 b ρ z r 2 0 1 - 1 2 b 0 1 2 b z ρ r 2 0 1 - 1 2 b ρ 2 r 2 ] G_{ik}=\frac{1}{4\pi\mu r}\begin{bmatrix}1-\frac{1}{2b}\frac{z^{2}}{r^{2}}&0&% \frac{1}{2b}\frac{\rho z}{r^{2}}\\ 0&1-\frac{1}{2b}&0\\ \frac{1}{2b}\frac{z\rho}{r^{2}}&0&1-\frac{1}{2b}\frac{\rho^{2}}{r^{2}}\end{% bmatrix}\,\!
  115. F z F_{z}\,\!
  116. ρ ^ \hat{\mathbf{\rho}}\,\!
  117. 𝐳 ^ \hat{\mathbf{z}}\,\!
  118. ρ \rho\,\!
  119. z z\,\!
  120. 𝐮 = F z 4 π μ r [ 1 4 ( 1 - ν ) ρ z r 2 ρ ^ + ( 1 - 1 4 ( 1 - ν ) ρ 2 r 2 ) 𝐳 ^ ] \mathbf{u}=\frac{F_{z}}{4\pi\mu r}\left[\frac{1}{4(1-\nu)}\,\frac{\rho z}{r^{2% }}\hat{\mathbf{\rho}}+\left(1-\frac{1}{4(1-\nu)}\,\frac{\rho^{2}}{r^{2}}\right% )\hat{\mathbf{z}}\right]\,\!
  121. G i k = 1 4 π μ [ b r + x 2 r 3 - a x 2 r ( r + z ) 2 - a z r ( r + z ) x y r 3 - a x y r ( r + z ) 2 x z r 3 - a x r ( r + z ) y x r 3 - a y x r ( r + z ) 2 b r + y 2 r 3 - a y 2 r ( r + z ) 2 - a z r ( r + z ) y z r 3 - a y r ( r + z ) z x r 3 + a x r ( r + z ) z y r 3 + a y r ( r + z ) b r + z 2 r 3 ] G_{ik}=\frac{1}{4\pi\mu}\begin{bmatrix}\frac{b}{r}+\frac{x^{2}}{r^{3}}-\frac{% ax^{2}}{r(r+z)^{2}}-\frac{az}{r(r+z)}&\frac{xy}{r^{3}}-\frac{axy}{r(r+z)^{2}}&% \frac{xz}{r^{3}}-\frac{ax}{r(r+z)}\\ \frac{yx}{r^{3}}-\frac{ayx}{r(r+z)^{2}}&\frac{b}{r}+\frac{y^{2}}{r^{3}}-\frac{% ay^{2}}{r(r+z)^{2}}-\frac{az}{r(r+z)}&\frac{yz}{r^{3}}-\frac{ay}{r(r+z)}\\ \frac{zx}{r^{3}}+\frac{ax}{r(r+z)}&\frac{zy}{r^{3}}+\frac{ay}{r(r+z)}&\frac{b}% {r}+\frac{z^{2}}{r^{3}}\end{bmatrix}\,\!
  122. σ j i , j + F i = ρ u ¨ i = ρ t t u i . \sigma_{ji,j}+F_{i}=\rho\,\ddot{u}_{i}=\rho\,\partial_{tt}u_{i}.\,\!
  123. μ u i , j j + ( μ + λ ) u j , i j + F i = ρ t t u i or μ 2 𝐮 + ( μ + λ ) ( 𝐮 ) + 𝐅 = ρ 2 𝐮 t 2 . \mu u_{i,jj}+(\mu+\lambda)u_{j,ij}+F_{i}=\rho\partial_{tt}u_{i}\quad\mathrm{or% }\quad\mu\nabla^{2}\mathbf{u}+(\mu+\lambda)\nabla(\nabla\cdot\mathbf{u})+% \mathbf{F}=\rho\frac{\partial^{2}\mathbf{u}}{\partial t^{2}}.\,\!
  124. ( δ k l t t - A k l [ ] ) u l = 1 ρ F k (\delta_{kl}\partial_{tt}-A_{kl}[\nabla])\,u_{l}=\frac{1}{\rho}F_{k}\,\!
  125. A k l [ ] = 1 ρ i C i k l j j A_{kl}[\nabla]=\frac{1}{\rho}\,\partial_{i}\,C_{iklj}\,\partial_{j}\,\!
  126. δ k l \delta_{kl}\,\!
  127. C i j k l = K δ i j δ k l + μ ( δ i k δ j l + δ i l δ j k - 2 3 δ i j δ k l ) C_{ijkl}=K\,\delta_{ij}\,\delta_{kl}+\mu\,(\delta_{ik}\delta_{jl}+\delta_{il}% \delta_{jk}-\frac{2}{3}\,\delta_{ij}\,\delta_{kl})\,\!
  128. K K\,\!
  129. μ \mu\,\!
  130. A i j [ ] = α 2 i j + β 2 ( m m δ i j - i j ) A_{ij}[\nabla]=\alpha^{2}\partial_{i}\partial_{j}+\beta^{2}(\partial_{m}% \partial_{m}\delta_{ij}-\partial_{i}\partial_{j})\,\!
  131. A i j [ 𝐤 ] = α 2 k i k j + β 2 ( k m k m δ i j - k i k j ) A_{ij}[\mathbf{k}]=\alpha^{2}k_{i}k_{j}+\beta^{2}(k_{m}k_{m}\delta_{ij}-k_{i}k% _{j})\,\!
  132. α 2 = ( K + 4 3 μ ) / ρ β 2 = μ / ρ \alpha^{2}=\left(K+\frac{4}{3}\mu\right)/\rho\qquad\beta^{2}=\mu/\rho\,\!
  133. A [ 𝐤 ^ ] A[\hat{\mathbf{k}}]\,\!
  134. 𝐮 ^ \hat{\mathbf{u}}\,\!
  135. 𝐤 ^ \hat{\mathbf{k}}\,\!
  136. C i j k l C_{ijkl}\,\!
  137. σ i j \sigma_{ij}\,\!
  138. ε i j \varepsilon_{ij}\,\!
  139. C i j k l C_{ijkl}\,\!
  140. C α β C_{\alpha\beta}\,\!
  141. i j = α = 11 22 33 23 , 32 13 , 31 12 , 21 1 2 3 4 5 6 \begin{matrix}ij&=\\ \Downarrow&\\ \alpha&=\end{matrix}\begin{matrix}11&22&33&23,32&13,31&12,21\\ \Downarrow&\Downarrow&\Downarrow&\Downarrow&\Downarrow&\Downarrow&\\ 1&2&3&4&5&6\end{matrix}\,\!
  142. C i j k l C α β = [ C 11 C 12 C 13 C 14 C 15 C 16 C 12 C 22 C 23 C 24 C 25 C 26 C 13 C 23 C 33 C 34 C 35 C 36 C 14 C 24 C 34 C 44 C 45 C 46 C 15 C 25 C 35 C 45 C 55 C 56 C 16 C 26 C 36 C 46 C 56 C 66 ] . C_{ijkl}\Rightarrow C_{\alpha\beta}=\begin{bmatrix}C_{11}&C_{12}&C_{13}&C_{14}% &C_{15}&C_{16}\\ C_{12}&C_{22}&C_{23}&C_{24}&C_{25}&C_{26}\\ C_{13}&C_{23}&C_{33}&C_{34}&C_{35}&C_{36}\\ C_{14}&C_{24}&C_{34}&C_{44}&C_{45}&C_{46}\\ C_{15}&C_{25}&C_{35}&C_{45}&C_{55}&C_{56}\\ C_{16}&C_{26}&C_{36}&C_{46}&C_{56}&C_{66}\end{bmatrix}.\,\!
  143. C α β C_{\alpha\beta}\,\!
  144. σ i j = W ε i j \sigma_{ij}=\frac{\partial W}{\partial\varepsilon_{ij}}
  145. C α β C_{\alpha\beta}\,\!
  146. C α β = [ K + 4 μ / 3 K - 2 μ / 3 K - 2 μ / 3 0 0 0 K - 2 μ / 3 K + 4 μ / 3 K - 2 μ / 3 0 0 0 K - 2 μ / 3 K - 2 μ / 3 K + 4 μ / 3 0 0 0 0 0 0 μ 0 0 0 0 0 0 μ 0 0 0 0 0 0 μ ] . C_{\alpha\beta}=\begin{bmatrix}K+4\mu\ /3&K-2\mu\ /3&K-2\mu\ /3&0&0&0\\ K-2\mu\ /3&K+4\mu\ /3&K-2\mu\ /3&0&0&0\\ K-2\mu\ /3&K-2\mu\ /3&K+4\mu\ /3&0&0&0\\ 0&0&0&\mu&0&0\\ 0&0&0&0&\mu&0\\ 0&0&0&0&0&\mu\end{bmatrix}.\,\!
  147. C α β = [ C 11 C 12 C 12 0 0 0 C 12 C 11 C 12 0 0 0 C 12 C 12 C 11 0 0 0 0 0 0 C 44 0 0 0 0 0 0 C 44 0 0 0 0 0 0 C 44 ] . C_{\alpha\beta}=\begin{bmatrix}C_{11}&C_{12}&C_{12}&0&0&0\\ C_{12}&C_{11}&C_{12}&0&0&0\\ C_{12}&C_{12}&C_{11}&0&0&0\\ 0&0&0&C_{44}&0&0\\ 0&0&0&0&C_{44}&0\\ 0&0&0&0&0&C_{44}\end{bmatrix}.\,\!
  148. C α β = [ C 11 C 11 - 2 C 66 C 13 0 0 0 C 11 - 2 C 66 C 11 C 13 0 0 0 C 13 C 13 C 33 0 0 0 0 0 0 C 44 0 0 0 0 0 0 C 44 0 0 0 0 0 0 C 66 ] . C_{\alpha\beta}=\begin{bmatrix}C_{11}&C_{11}-2C_{66}&C_{13}&0&0&0\\ C_{11}-2C_{66}&C_{11}&C_{13}&0&0&0\\ C_{13}&C_{13}&C_{33}&0&0&0\\ 0&0&0&C_{44}&0&0\\ 0&0&0&0&C_{44}&0\\ 0&0&0&0&0&C_{66}\end{bmatrix}.\,\!
  149. C α β = [ C 11 C 12 C 13 0 0 0 C 12 C 22 C 23 0 0 0 C 13 C 23 C 33 0 0 0 0 0 0 C 44 0 0 0 0 0 0 C 55 0 0 0 0 0 0 C 66 ] . C_{\alpha\beta}=\begin{bmatrix}C_{11}&C_{12}&C_{13}&0&0&0\\ C_{12}&C_{22}&C_{23}&0&0&0\\ C_{13}&C_{23}&C_{33}&0&0&0\\ 0&0&0&C_{44}&0&0\\ 0&0&0&0&C_{55}&0\\ 0&0&0&0&0&C_{66}\end{bmatrix}.\,\!
  150. ( δ k l t t - A k l [ ] ) u l = 1 ρ F k (\delta_{kl}\partial_{tt}-A_{kl}[\nabla])\,u_{l}=\frac{1}{\rho}F_{k}\,\!
  151. A k l [ ] = 1 ρ i C i k l j j A_{kl}[\nabla]=\frac{1}{\rho}\,\partial_{i}\,C_{iklj}\,\partial_{j}\,\!
  152. δ k l \delta_{kl}\,\!
  153. 𝐮 [ 𝐱 , t ] = U [ 𝐤 𝐱 - ω t ] 𝐮 ^ \mathbf{u}[\mathbf{x},\,t]=U[\mathbf{k}\cdot\mathbf{x}-\omega\,t]\,\hat{% \mathbf{u}}\,\!
  154. 𝐮 ^ \hat{\mathbf{u}}\,\!
  155. ω 2 \omega^{2}\,\!
  156. 𝐮 ^ \hat{\mathbf{u}}\,\!
  157. A k l [ 𝐤 ] = 1 ρ k i C i k l j k j . A_{kl}[\mathbf{k}]=\frac{1}{\rho}\,k_{i}\,C_{iklj}\,k_{j}.\,\!
  158. A [ 𝐤 ^ ] 𝐮 ^ = c 2 𝐮 ^ A[\hat{\mathbf{k}}]\,\hat{\mathbf{u}}=c^{2}\,\hat{\mathbf{u}}\,\!
  159. 𝐤 ^ = 𝐤 / 𝐤 𝐤 \hat{\mathbf{k}}=\mathbf{k}/\sqrt{\mathbf{k}\cdot\mathbf{k}}\,\!
  160. c = ω / 𝐤 𝐤 c=\omega/\sqrt{\mathbf{k}\cdot\mathbf{k}}\,\!

Linear_form.html

  1. f ( v + w ) = f ( v ) + f ( w ) f(\vec{v}+\vec{w})=f(\vec{v})+f(\vec{w})
  2. v , w V \vec{v},\vec{w}\in V
  3. f ( a v ) = a f ( v ) f(a\vec{v})=af(\vec{v})
  4. v V , a k . \vec{v}\in V,a\in k.
  5. x = [ x 1 x n ] . \vec{x}=\begin{bmatrix}x_{1}\\ \vdots\\ x_{n}\end{bmatrix}.
  6. f ( x ) = a 1 x 1 + + a n x n . f(\vec{x})=a_{1}x_{1}+\cdots+a_{n}x_{n}.
  7. x \vec{x}
  8. f ( x ) = [ a 1 a n ] [ x 1 x n ] . f(\vec{x})=[a_{1}\dots a_{n}]\begin{bmatrix}x_{1}\\ \vdots\\ x_{n}\end{bmatrix}.
  9. I ( f ) = a b f ( x ) d x I(f)=\int_{a}^{b}f(x)\,dx
  10. I ( f + g ) = a b ( f ( x ) + g ( x ) ) d x = a b f ( x ) d x + a b g ( x ) d x = I ( f ) + I ( g ) I(f+g)=\int_{a}^{b}(f(x)+g(x))\,dx=\int_{a}^{b}f(x)\,dx+\int_{a}^{b}g(x)\,dx=I% (f)+I(g)
  11. I ( α f ) = a b α f ( x ) d x = α a b f ( x ) d x = α I ( f ) . I(\alpha f)=\int_{a}^{b}\alpha f(x)\,dx=\alpha\int_{a}^{b}f(x)\,dx=\alpha I(f).
  12. ev c f = f ( c ) . \operatorname{ev}_{c}f=f(c).
  13. ( f + g ) ( c ) = f ( c ) + g ( c ) (f+g)(c)=f(c)+g(c)
  14. ( α f ) ( c ) = α f ( c ) . (\alpha f)(c)=\alpha f(c).
  15. I ( f ) = a 0 f ( x 0 ) + a 1 f ( x 1 ) + + a n f ( x n ) I(f)=a_{0}f(x_{0})+a_{1}f(x_{1})+\dots+a_{n}f(x_{n})
  16. V V * : v v * V\to V^{*}:v\mapsto v^{*}
  17. v * ( w ) := v , w . v^{*}(w):=\langle v,w\rangle.
  18. V * V : f f * V^{*}\to V:f\mapsto f^{*}
  19. f * , w = f ( w ) . \langle f^{*},w\rangle=f(w).
  20. e 1 , e 2 , , e n \vec{e}_{1},\vec{e}_{2},\dots,\vec{e}_{n}
  21. ω ~ 1 , ω ~ 2 , , ω ~ n \tilde{\omega}^{1},\tilde{\omega}^{2},\dots,\tilde{\omega}^{n}
  22. ω ~ i ( e j ) = { 1 if i = j 0 if i j . \tilde{\omega}^{i}(\vec{e}_{j})=\left\{\begin{matrix}1&\mathrm{if}\ i=j\\ 0&\mathrm{if}\ i\not=j.\end{matrix}\right.
  23. ω ~ i ( e j ) = δ j i \tilde{\omega}^{i}(\vec{e}_{j})=\delta^{i}_{j}
  24. u ~ \tilde{u}
  25. V ~ \tilde{V}
  26. u ~ = i = 1 n u i ω ~ i . \tilde{u}=\sum_{i=1}^{n}u_{i}\,\tilde{\omega}^{i}.
  27. u ~ \tilde{u}
  28. u ~ ( e j ) = i = 1 n ( u i ω ~ i ) e j = i u i ( ω ~ i ( e j ) ) \tilde{u}(\vec{e}_{j})=\sum_{i=1}^{n}(u_{i}\,\tilde{\omega}^{i})\vec{e}_{j}=% \sum_{i}u_{i}(\tilde{\omega}^{i}(\vec{e}_{j}))
  29. u ~ ( e j ) = i u i ( ω ~ i ( e j ) ) = i u i δ i = j u j \tilde{u}({\vec{e}}_{j})=\sum_{i}u_{i}(\tilde{\omega}^{i}({\vec{e}}_{j}))=\sum% _{i}u_{i}\delta^{i}{}_{j}=u_{j}
  30. u ~ ( e j ) = u j . \tilde{u}(\vec{e}_{j})=u_{j}.
  31. e 1 , , e n \vec{e}_{1},\dots,\vec{e}_{n}
  32. ω ~ i ( v ) = 1 2 j = 1 3 k = 1 3 ε i j k ( e j × e k ) e 1 e 2 × e 3 , v . \tilde{\omega}^{i}(\vec{v})={1\over 2}\,\left\langle{\sum_{j=1}^{3}\sum_{k=1}^% {3}\varepsilon^{ijk}\,(\vec{e}_{j}\times\vec{e}_{k})\over\vec{e}_{1}\cdot\vec{% e}_{2}\times\vec{e}_{3}},\vec{v}\right\rangle.
  33. , \langle,\rangle
  34. ω ~ i ( v ) = 1 i 2 < i 3 < < i n n ε i i 2 i n ( e i 2 e i n ) ( e 1 e n ) , v \tilde{\omega}^{i}(\vec{v})=\left\langle\frac{\underset{{}^{1\leq i_{2}<i_{3}<% \dots<i_{n}\leq n}}{\sum}\varepsilon^{ii_{2}\dots i_{n}}(\star\vec{e}_{i_{2}}% \wedge\dots\wedge\vec{e}_{i_{n}})}{\star(\vec{e}_{1}\wedge\dots\wedge\vec{e}_{% n})},\vec{v}\right\rangle
  35. \star

Linear_function.html

  1. f ( x ) = a x + b , f(x)=ax+b,
  2. a a
  3. b b
  4. a a
  5. b b
  6. f ( x 1 , , x k ) f(x_{1},\ldots,x_{k})
  7. f ( x 1 , , x k ) = b + a 1 x 1 + + a k x k f(x_{1},\ldots,x_{k})=b+a_{1}x_{1}+\ldots+a_{k}x_{k}
  8. f ( 𝐱 + 𝐲 ) = f ( 𝐱 ) + f ( 𝐲 ) f(\mathbf{x}+\mathbf{y})=f(\mathbf{x})+f(\mathbf{y})
  9. f ( a 𝐱 ) = a f ( 𝐱 ) . f(a\mathbf{x})=af(\mathbf{x}).
  10. a a
  11. K K
  12. 𝐱 \mathbf{x}
  13. 𝐲 \mathbf{y}
  14. K K
  15. f ( 0 [ , , 0 ] ) = 0 f(0[,\ldots,0])=0
  16. b = 0 b=0

Linear_independence.html

  1. a 1 v 1 + a 2 v 2 + + a k v k = 0 , a_{1}v_{1}+a_{2}v_{2}+\cdots+a_{k}v_{k}=0,
  2. v 1 = - a 2 a 1 v 2 + + - a k a 1 v k . v_{1}=\frac{-a_{2}}{a_{1}}v_{2}+\cdots+\frac{-a_{k}}{a_{1}}v_{k}.
  3. a 1 v 1 + a 2 v 2 + + a n v n = 0 , a_{1}v_{1}+a_{2}v_{2}+\cdots+a_{n}v_{n}=0,
  4. j J a j v j = 0 \sum_{j\in J}a_{j}v_{j}=0\,
  5. a 1 { 1 1 } + a 2 { - 3 2 } + a 3 { 2 4 } = { 0 0 } , a_{1}\begin{Bmatrix}1\\ 1\end{Bmatrix}+a_{2}\begin{Bmatrix}-3\\ 2\end{Bmatrix}+a_{3}\begin{Bmatrix}2\\ 4\end{Bmatrix}=\begin{Bmatrix}0\\ 0\end{Bmatrix},
  6. [ 1 - 3 2 1 2 4 ] { a 1 a 2 a 3 } = { 0 0 } . \begin{bmatrix}1&-3&2\\ 1&2&4\end{bmatrix}\begin{Bmatrix}a_{1}\\ a_{2}\\ a_{3}\end{Bmatrix}=\begin{Bmatrix}0\\ 0\end{Bmatrix}.
  7. [ 1 - 3 2 0 5 2 ] { a 1 a 2 a 3 } = { 0 0 } . \begin{bmatrix}1&-3&2\\ 0&5&2\end{bmatrix}\begin{Bmatrix}a_{1}\\ a_{2}\\ a_{3}\end{Bmatrix}=\begin{Bmatrix}0\\ 0\end{Bmatrix}.
  8. [ 1 0 16 / 5 0 1 2 / 5 ] { a 1 a 2 a 3 } = { 0 0 } . \begin{bmatrix}1&0&16/5\\ 0&1&2/5\end{bmatrix}\begin{Bmatrix}a_{1}\\ a_{2}\\ a_{3}\end{Bmatrix}=\begin{Bmatrix}0\\ 0\end{Bmatrix}.
  9. [ 1 0 0 1 ] { a 1 a 2 } = { a 1 a 2 } = - a 3 { 16 / 5 2 / 5 } . \begin{bmatrix}1&0\\ 0&1\end{bmatrix}\begin{Bmatrix}a_{1}\\ a_{2}\end{Bmatrix}=\begin{Bmatrix}a_{1}\\ a_{2}\end{Bmatrix}=-a_{3}\begin{Bmatrix}16/5\\ 2/5\end{Bmatrix}.
  10. a 1 { 1 1 } + a 2 { - 3 2 } = { 0 0 } , a_{1}\begin{Bmatrix}1\\ 1\end{Bmatrix}+a_{2}\begin{Bmatrix}-3\\ 2\end{Bmatrix}=\begin{Bmatrix}0\\ 0\end{Bmatrix},
  11. [ 1 - 3 1 2 ] { a 1 a 2 } = { 0 0 } . \begin{bmatrix}1&-3\\ 1&2\end{bmatrix}\begin{Bmatrix}a_{1}\\ a_{2}\end{Bmatrix}=\begin{Bmatrix}0\\ 0\end{Bmatrix}.
  12. [ 1 0 0 1 ] { a 1 a 2 } = { 0 0 } . \begin{bmatrix}1&0\\ 0&1\end{bmatrix}\begin{Bmatrix}a_{1}\\ a_{2}\end{Bmatrix}=\begin{Bmatrix}0\\ 0\end{Bmatrix}.
  13. 𝐯 1 = { 1 4 2 - 3 } , 𝐯 2 = { 7 10 - 4 - 1 } , 𝐯 3 = { - 2 1 5 - 4 } . \mathbf{v}_{1}=\begin{Bmatrix}1\\ 4\\ 2\\ -3\end{Bmatrix},\mathbf{v}_{2}=\begin{Bmatrix}7\\ 10\\ -4\\ -1\end{Bmatrix},\mathbf{v}_{3}=\begin{Bmatrix}-2\\ 1\\ 5\\ -4\end{Bmatrix}.
  14. [ 1 7 - 2 4 10 1 2 - 4 5 - 3 - 1 - 4 ] { a 1 a 2 a 3 } = { 0 0 0 0 } . \begin{bmatrix}1&7&-2\\ 4&10&1\\ 2&-4&5\\ -3&-1&-4\end{bmatrix}\begin{Bmatrix}a_{1}\\ a_{2}\\ a_{3}\end{Bmatrix}=\begin{Bmatrix}0\\ 0\\ 0\\ 0\end{Bmatrix}.
  15. [ 1 7 - 2 0 - 18 9 0 0 0 0 0 0 ] { a 1 a 2 a 3 } = { 0 0 0 0 } . \begin{bmatrix}1&7&-2\\ 0&-18&9\\ 0&0&0\\ 0&0&0\end{bmatrix}\begin{Bmatrix}a_{1}\\ a_{2}\\ a_{3}\end{Bmatrix}=\begin{Bmatrix}0\\ 0\\ 0\\ 0\end{Bmatrix}.
  16. [ 1 7 0 - 18 ] { a 1 a 2 } = - a 3 { - 2 9 } . \begin{bmatrix}1&7\\ 0&-18&\end{bmatrix}\begin{Bmatrix}a_{1}\\ a_{2}\end{Bmatrix}=-a_{3}\begin{Bmatrix}-2\\ 9\end{Bmatrix}.
  17. a 1 = - 3 a 3 / 2 , a 2 = a 3 / 2 , a_{1}=-3a_{3}/2,a_{2}=a_{3}/2,
  18. n \mathbb{R}^{n}
  19. A = [ 1 - 3 1 2 ] . A=\begin{bmatrix}1&-3\\ 1&2\end{bmatrix}.\,\!
  20. A Λ = [ 1 - 3 1 2 ] [ λ 1 λ 2 ] . A\Lambda=\begin{bmatrix}1&-3\\ 1&2\end{bmatrix}\begin{bmatrix}\lambda_{1}\\ \lambda_{2}\end{bmatrix}.\,\!
  21. det A = 1 2 - 1 ( - 3 ) = 5 0. \det A=1\cdot 2-1\cdot(-3)=5\neq 0.\,\!
  22. A i 1 , , i m Λ = 0. A_{{\langle i_{1},\dots,i_{m}}\rangle}\Lambda={0}.\,\!
  23. det A i 1 , , i m = 0 \det A_{{\langle i_{1},\dots,i_{m}}\rangle}=0\,\!
  24. 𝐞 1 = ( 1 , 0 , 0 , , 0 ) 𝐞 2 = ( 0 , 1 , 0 , , 0 ) 𝐞 n = ( 0 , 0 , 0 , , 1 ) . \begin{matrix}\mathbf{e}_{1}&=&(1,0,0,\ldots,0)\\ \mathbf{e}_{2}&=&(0,1,0,\ldots,0)\\ &\vdots\\ \mathbf{e}_{n}&=&(0,0,0,\ldots,1).\end{matrix}
  25. a 1 𝐞 1 + a 2 𝐞 2 + + a n 𝐞 n = 0. a_{1}\mathbf{e}_{1}+a_{2}\mathbf{e}_{2}+\cdots+a_{n}\mathbf{e}_{n}=0.\,\!
  26. a 1 𝐞 1 + a 2 𝐞 2 + + a n 𝐞 n = ( a 1 , a 2 , , a n ) , a_{1}\mathbf{e}_{1}+a_{2}\mathbf{e}_{2}+\cdots+a_{n}\mathbf{e}_{n}=(a_{1},a_{2% },\ldots,a_{n}),\,\!
  27. a 1 𝐯 1 + + a n 𝐯 n = 0. a_{1}\mathbf{v}_{1}+\cdots+a_{n}\mathbf{v}_{n}=0.\,

Linear_interpolation.html

  1. ( x 0 , y 0 ) (x_{0},y_{0})
  2. ( x 1 , y 1 ) (x_{1},y_{1})
  3. ( x 0 , x 1 ) (x_{0},x_{1})
  4. y - y 0 x - x 0 = y 1 - y 0 x 1 - x 0 \frac{y-y_{0}}{x-x_{0}}=\frac{y_{1}-y_{0}}{x_{1}-x_{0}}
  5. y = y 0 + ( y 1 - y 0 ) x - x 0 x 1 - x 0 y=y_{0}+(y_{1}-y_{0})\frac{x-x_{0}}{x_{1}-x_{0}}
  6. ( x 0 , x 1 ) (x_{0},x_{1})
  7. x - x 0 x 1 - x 0 \frac{x-x_{0}}{x_{1}-x_{0}}
  8. x 1 - x x 1 - x 0 \frac{x_{1}-x}{x_{1}-x_{0}}
  9. C 0 C^{0}
  10. R T = f ( x ) - p ( x ) R_{T}=f(x)-p(x)\,\!
  11. p ( x ) = f ( x 0 ) + f ( x 1 ) - f ( x 0 ) x 1 - x 0 ( x - x 0 ) . p(x)=f(x_{0})+\frac{f(x_{1})-f(x_{0})}{x_{1}-x_{0}}(x-x_{0}).\,\!
  12. | R T | ( x 1 - x 0 ) 2 8 max x 0 x x 1 | f ′′ ( x ) | . |R_{T}|\leq\frac{(x_{1}-x_{0})^{2}}{8}\max_{x_{0}\leq x\leq x_{1}}|f^{\prime% \prime}(x)|.\,\!

Linear_regulator.html

  1. I Z I_{\mathrm{Z}}
  2. R 1 = V S - V Z I Z + I R2 R1=\frac{V_{\mathrm{S}}-V_{\mathrm{Z}}}{I_{\mathrm{Z}}+I_{\mathrm{R2}}}
  3. V Z V_{\mathrm{Z}}
  4. V S V_{\mathrm{S}}
  5. R 1 = V S - V Z I Z + K I B R1=\frac{V_{\mathrm{S}}-V_{\mathrm{Z}}}{I_{\mathrm{Z}}+K\cdot I_{\mathrm{B}}}
  6. I B = I R2 h FE ( min ) I_{\mathrm{B}}=\frac{I_{\mathrm{R2}}}{h_{\mathrm{FE(min)}}}

Linearity.html

  1. f ( x ) = m x + b f(x)=mx+b
  2. f f
  3. a 0 , a 1 , , a n { 0 , 1 } a_{0},a_{1},\ldots,a_{n}\in\{0,1\}
  4. f ( b 1 , , b n ) = a 0 ( a 1 b 1 ) ( a n b n ) f(b_{1},\ldots,b_{n})=a_{0}\oplus(a_{1}\land b_{1})\oplus\cdots\oplus(a_{n}% \land b_{n})
  5. b 1 , , b n { 0 , 1 } . b_{1},\ldots,b_{n}\in\{0,1\}.

Linearity_of_differentiation.html

  1. f f
  2. g g
  3. α α
  4. β β
  5. d d x ( α f ( x ) + β g ( x ) ) \frac{\mbox{d}~{}}{\mbox{d}~{}x}(\alpha\cdot f(x)+\beta\cdot g(x))
  6. d d x ( α f ( x ) ) + d d x ( β g ( x ) ) \frac{\mbox{d}~{}}{\mbox{d}~{}x}(\alpha\cdot f(x))+\frac{\mbox{d}~{}}{\mbox{d}% ~{}x}(\beta\cdot g(x))
  7. α f ( x ) + β g ( x ) \alpha\cdot f^{\prime}(x)+\beta\cdot g^{\prime}(x)
  8. d d x ( α f ( x ) + β g ( x ) ) = α f ( x ) + β g ( x ) \frac{\mbox{d}~{}}{\mbox{d}~{}x}(\alpha\cdot f(x)+\beta\cdot g(x))=\alpha\cdot f% ^{\prime}(x)+\beta\cdot g^{\prime}(x)
  9. ( α f + β g ) = α f + β g (\alpha\cdot f+\beta\cdot g)^{\prime}=\alpha\cdot f^{\prime}+\beta\cdot g^{\prime}

Linearity_of_integration.html

  1. f f
  2. g g
  3. ( a f ( x ) + b g ( x ) ) d x . \int(af(x)+bg(x))\,dx.
  4. a f ( x ) d x + b g ( x ) d x . \int af(x)\,dx+\int bg(x)\,dx.
  5. a f ( x ) d x + b g ( x ) d x . a\int f(x)\,dx+b\int g(x)\,dx.
  6. ( a f ( x ) + b g ( x ) ) d x = a f ( x ) d x + b g ( x ) d x . \int(af(x)+bg(x))\,dx=a\int f(x)\,dx+b\int g(x)\,dx.

Linnik's_theorem.html

  1. a + n d , a+nd,
  2. p ( a , d ) < c d L . p(a,d)<cd^{L}.\;
  3. p ( a , d ) ( 1 + o ( 1 ) ) φ ( d ) 2 ln 2 d , p(a,d)\leq(1+o(1))\varphi(d)^{2}\ln^{2}d\;,
  4. φ \varphi
  5. p ( a , d ) < d 2 . p(a,d)<d^{2}.\;

Liouville's_theorem_(complex_analysis).html

  1. | f ( z ) | M |f(z)|\leq M
  2. z z
  3. \mathbb{C}
  4. \mathbb{C}
  5. f ( z ) = k = 0 a k z k f(z)=\sum_{k=0}^{\infty}a_{k}z^{k}
  6. a k = f ( k ) ( 0 ) k ! = 1 2 π i C r f ( ζ ) ζ k + 1 d ζ a_{k}=\frac{f^{(k)}(0)}{k!}={1\over 2\pi i}\oint_{C_{r}}\frac{f(\zeta)}{\zeta^% {k+1}}\,d\zeta
  7. | a k | 1 2 π C r | f ( ζ ) | | ζ | k + 1 | d ζ | 1 2 π C r M r k + 1 | d ζ | = M 2 π r k + 1 C r | d ζ | = M 2 π r k + 1 2 π r = M r k , |a_{k}|\leq\frac{1}{2\pi}\oint_{C_{r}}\frac{|f(\zeta)|}{|\zeta|^{k+1}}\,|d% \zeta|\leq\frac{1}{2\pi}\oint_{C_{r}}\frac{M}{r^{k+1}}\,|d\zeta|=\frac{M}{2\pi r% ^{k+1}}\oint_{C_{r}}|d\zeta|=\frac{M}{2\pi r^{k+1}}2\pi r=\frac{M}{r^{k}},
  8. | f ( z ) | = 1 2 π | C r f ( ζ ) ( ζ - z ) 2 d ζ | 1 2 π C r | f ( ζ ) | | ( ζ - z ) 2 | | d ζ | 1 2 π C r M | ζ | | ( ζ - z ) 2 | | d ζ | = M I 2 π |f^{\prime}(z)|=\frac{1}{2\pi}\left|\oint_{C_{r}}\frac{f(\zeta)}{(\zeta-z)^{2}% }d\zeta\right|\leq\frac{1}{2\pi}\oint_{C_{r}}\frac{\left|f(\zeta)\right|}{% \left|(\zeta-z)^{2}\right|}\left|d\zeta\right|\leq\frac{1}{2\pi}\oint_{C_{r}}% \frac{M\left|\zeta\right|}{\left|(\zeta-z)^{2}\right|}\left|d\zeta\right|=% \frac{MI}{2\pi}
  9. ( z ) : | g ( z ) | = 1 | f ( z ) - w | < 1 r (\forall z\in\mathbb{C}):|g(z)|=\frac{1}{|f(z)-w|}<\frac{1}{r}\cdot
  10. f ( z ) = k = 0 a k z k . f(z)=\sum_{k=0}^{\infty}a_{k}z^{k}.
  11. ( k ) : | a k | M r n - k . (\forall k\in\mathbb{N}):|a_{k}|\leqslant Mr^{n-k}.
  12. | a k | lim r + M r n - k = 0. |a_{k}|\leqslant\lim_{r\rightarrow+\infty}Mr^{n-k}=0.

Liouville's_theorem_(Hamiltonian).html

  1. q i q_{i}
  2. p i p_{i}
  3. i = 1 , , n i=1,\dots,n
  4. ρ ( p , q ) \rho(p,q)
  5. ρ ( p , q ) d n q d n p \rho(p,q)\,d^{n}q\,d^{n}p
  6. d n q d n p d^{n}q\,d^{n}p
  7. ρ ( p , q ; t ) \rho(p,q;t)
  8. t t
  9. d ρ d t = ρ t + i = 1 n ( ρ q i q ˙ i + ρ p i p ˙ i ) = 0. \frac{d\rho}{dt}=\frac{\partial\rho}{\partial t}+\sum_{i=1}^{n}\left(\frac{% \partial\rho}{\partial q_{i}}\dot{q}_{i}+\frac{\partial\rho}{\partial p_{i}}% \dot{p}_{i}\right)=0.
  10. ρ \rho
  11. ρ t + i = 1 n ( ( ρ q ˙ i ) q i + ( ρ p ˙ i ) p i ) = 0. \frac{\partial\rho}{\partial t}+\sum_{i=1}^{n}\left(\frac{\partial(\rho\dot{q}% _{i})}{\partial q_{i}}+\frac{\partial(\rho\dot{p}_{i})}{\partial p_{i}}\right)% =0.
  12. ( ρ , ρ q ˙ i , ρ p ˙ i ) (\rho,\rho\dot{q}_{i},\rho\dot{p}_{i})
  13. ρ i = 1 n ( q ˙ i q i + p ˙ i p i ) = ρ i = 1 n ( 2 H q i p i - 2 H p i q i ) = 0 , \rho\sum_{i=1}^{n}\left(\frac{\partial\dot{q}_{i}}{\partial q_{i}}+\frac{% \partial\dot{p}_{i}}{\partial p_{i}}\right)=\rho\sum_{i=1}^{n}\left(\frac{% \partial^{2}H}{\partial q_{i}\,\partial p_{i}}-\frac{\partial^{2}H}{\partial p% _{i}\partial q_{i}}\right)=0,
  14. H H
  15. d ρ / d t d\rho/dt
  16. ( p ˙ , q ˙ ) (\dot{p},\dot{q})
  17. p i p_{i}
  18. q i q^{i}
  19. Δ p i Δ q i \Delta p_{i}\,\Delta q^{i}
  20. ρ t = - { ρ , H } \frac{\partial\rho}{\partial t}=-\{\,\rho,H\,\}
  21. i 𝐋 ^ = i = 1 n [ H p i q i - H q i p i ] = { , H } \mathrm{i}\hat{\mathbf{L}}=\sum_{i=1}^{n}\left[\frac{\partial H}{\partial p_{i% }}\frac{\partial}{\partial q^{i}}-\frac{\partial H}{\partial q^{i}}\frac{% \partial}{\partial p_{i}}\right]=\{\cdot,H\}
  22. ρ t + i 𝐋 ^ ρ = 0. \frac{\partial\rho}{\partial t}+{\mathrm{i}\hat{\mathbf{L}}}\rho=0.
  23. ρ t = 1 i [ H , ρ ] \frac{\partial\rho}{\partial t}=\frac{1}{i\hbar}[H,\rho]
  24. d d t A = 1 i [ A , H ] \frac{d}{dt}\langle A\rangle=\frac{1}{i\hbar}\langle[A,H]\rangle
  25. A A

Liouville_function.html

  1. λ ( n ) = ( - 1 ) Ω ( n ) , \lambda(n)=(-1)^{\Omega(n)},\,\!
  2. d | n λ ( d ) = { 1 if n is a perfect square, 0 otherwise. \sum_{d|n}\lambda(d)=\begin{cases}1&\,\text{if }n\,\text{ is a perfect square,% }\\ 0&\,\text{otherwise.}\end{cases}
  3. ζ ( 2 s ) ζ ( s ) = n = 1 λ ( n ) n s . \frac{\zeta(2s)}{\zeta(s)}=\sum_{n=1}^{\infty}\frac{\lambda(n)}{n^{s}}.
  4. n = 1 λ ( n ) q n 1 - q n = n = 1 q n 2 = 1 2 ( ϑ 3 ( q ) - 1 ) , \sum_{n=1}^{\infty}\frac{\lambda(n)q^{n}}{1-q^{n}}=\sum_{n=1}^{\infty}q^{n^{2}% }=\frac{1}{2}\left(\vartheta_{3}(q)-1\right),
  5. ϑ 3 ( q ) \vartheta_{3}(q)
  6. L ( n ) = k = 1 n λ ( k ) , L(n)=\sum_{k=1}^{n}\lambda(k),
  7. L ( n ) 0 L(n)\leq 0

Liouville–Neumann_series.html

  1. ϕ ( x ) = n = 0 λ n ϕ n ( x ) \phi\left(x\right)=\sum^{\infty}_{n=0}\lambda^{n}\phi_{n}\left(x\right)
  2. f ( t ) = ϕ ( t ) - λ a b K ( t , s ) ϕ ( s ) d s . f(t)=\phi(t)-\lambda\int_{a}^{b}K(t,s)\phi(s)\,ds.
  3. K n ( x , z ) = K ( x , y 1 ) K ( y 1 , y 2 ) K ( y n - 1 , z ) d y 1 d y 2 d y n - 1 K_{n}\left(x,z\right)=\int\int\cdots\int K\left(x,y_{1}\right)K\left(y_{1},y_{% 2}\right)\cdots K\left(y_{n-1},z\right)dy_{1}dy_{2}\cdots dy_{n-1}
  4. ϕ n ( x ) = K n ( x , z ) f ( z ) d z \phi_{n}\left(x\right)=\int K_{n}\left(x,z\right)f\left(z\right)dz
  5. ϕ 0 ( x ) = f ( x ) . \phi_{0}\left(x\right)=f\left(x\right).
  6. K ( x , z ; λ ) = n = 0 λ n K n + 1 ( x , z ) . K\left(x,z;\lambda\right)=\sum^{\infty}_{n=0}\lambda^{n}K_{n+1}\left(x,z\right).
  7. ϕ ( x ) = K ( x , z ; λ ) f ( z ) d z . \phi\left(x\right)=\int K\left(x,z;\lambda\right)f\left(z\right)dz.

Lipid_bilayer.html

  1. Λ \Lambda
  2. Λ \Lambda

Liquid_air_cycle_engine.html

  1. 1 / ( 1 + 1 / ( ( L / D ) ( a / g ) ) ) 1/(1+1/((L/D)(a/g)))
  2. ( L / D ) (L/D)
  3. ( a / g ) (a/g)

Liquid_water_path.html

  1. L W P = z = 0 ρ a i r r L d z LWP=\int_{z=0}^{\infty}\rho_{air}r_{L}dz^{\prime}
  2. d p d z = - ρ a i r g \frac{dp}{dz}=-\rho_{air}g
  3. L W P = 0 p = p 0 r L d p / g LWP=\int_{0}^{p=p_{0}}r_{L}dp/g
  4. < v a r > g <var>g

Liskov_substitution_principle.html

  1. Φ ( x ) Φ(x)
  2. x x
  3. T . T.
  4. Φ ( y ) Φ(y)
  5. y y
  6. S S
  7. S S
  8. T . T.

List_(abstract_data_type).html

  1. return : A A * = a cons a nil \,\text{return}\colon A\to A^{*}=a\mapsto\,\text{cons}\,a\,\,\text{nil}
  2. bind : A * ( A B * ) B * = l f { nil if l = nil append ( f a ) ( bind l f ) if l = cons a l \,\text{bind}\colon A^{*}\to(A\to B^{*})\to B^{*}=l\mapsto f\mapsto\begin{% cases}\,\text{nil}&\,\text{if}\ l=\,\text{nil}\\ \,\text{append}\,(f\,a)\,(\,\text{bind}\,l^{\prime}\,f)&\,\text{if}\ l=\,\text% {cons}\,a\,l^{\prime}\end{cases}
  3. append : A * A * A * = l 1 l 2 { l 2 if l 1 = nil cons a ( append l 1 l 2 ) if l 1 = cons a l 1 \,\text{append}\colon A^{*}\to A^{*}\to A^{*}=l_{1}\mapsto l_{2}\mapsto\begin{% cases}l_{2}&\,\text{if}\ l_{1}=\,\text{nil}\\ \,\text{cons}\,a\,(\,\text{append}\,l_{1}^{\prime}\,l_{2})&\,\text{if}\ l_{1}=% \,\text{cons}\,a\,l_{1}^{\prime}\end{cases}
  4. fmap : ( A B ) ( A * B * ) = f l { nil if l = nil cons ( f a ) ( fmap f l ) if l = cons a l \,\text{fmap}\colon(A\to B)\to(A^{*}\to B^{*})=f\mapsto l\mapsto\begin{cases}% \,\text{nil}&\,\text{if}\ l=\,\text{nil}\\ \,\text{cons}\,(f\,a)(\,\text{fmap}f\,l^{\prime})&\,\text{if}\ l=\,\text{cons}% \,a\,l^{\prime}\end{cases}
  5. join : A * * A * = l { nil if l = nil append a ( join l ) if l = cons a l \,\text{join}\colon{A^{*}}^{*}\to A^{*}=l\mapsto\begin{cases}\,\text{nil}&\,% \text{if}\ l=\,\text{nil}\\ \,\text{append}\,a\,(\,\text{join}\,l^{\prime})&\,\text{if}\ l=\,\text{cons}\,% a\,l^{\prime}\end{cases}

List_comprehension.html

  1. S = { 2 x x , x 2 > 3 } S=\{\,2\cdot x\mid x\in\mathbb{N},\ x^{2}>3\,\}
  2. S S
  3. x x
  4. x x
  5. \mathbb{N}
  6. x x
  7. 3 3
  8. S = { 2 x \color V i o l e t output expression x \color V i o l e t variable \color V i o l e t input set , x 2 > 3 \color V i o l e t predicate } S=\{\,\underbrace{2\cdot x}_{\color{Violet}{\,\text{output expression}}}\mid% \underbrace{x}_{\color{Violet}{\,\text{variable}}}\in\underbrace{\mathbb{N}}_{% \color{Violet}{\,\text{input set}}},\ \underbrace{x^{2}>3}_{\color{Violet}{\,% \text{predicate}}}\,\}
  9. x x
  10. \mathbb{N}
  11. x 2 > 3 x^{2}>3
  12. 2 x 2\cdot x
  13. { } \{\}
  14. \mid
  15. , ,
  16. \mathbb{N}
  17. { 0 , 1 , , 100 } \{0,1,...,100\}
  18. \mathbb{N}

List_of_German_expressions_in_English.html

  1. \mathbb{Z}
  2. 𝕂 \mathbb{K}
  3. \mathbb{R}
  4. \mathbb{C}

List_of_integrals_of_exponential_functions.html

  1. f ( x ) e f ( x ) d x = e f ( x ) \int\mathrm{f}^{\prime}(x)e^{f(x)}\;\mathrm{d}x=e^{f(x)}
  2. e c x d x = 1 c e c x \int e^{cx}\;\mathrm{d}x=\frac{1}{c}e^{cx}
  3. a c x d x = 1 c ln a a c x \int a^{cx}\;\mathrm{d}x=\frac{1}{c\cdot\ln a}a^{cx}
  4. a > 0 , a 1 a>0,\ a\neq 1
  5. x e c x d x = e c x c 2 ( c x - 1 ) \int xe^{cx}\;\mathrm{d}x=\frac{e^{cx}}{c^{2}}(cx-1)
  6. x 2 e c x d x = e c x ( x 2 c - 2 x c 2 + 2 c 3 ) \int x^{2}e^{cx}\;\mathrm{d}x=e^{cx}\left(\frac{x^{2}}{c}-\frac{2x}{c^{2}}+% \frac{2}{c^{3}}\right)
  7. x n e c x d x = 1 c x n e c x - n c x n - 1 e c x d x = ( c ) n e c x c = e c x i = 0 n ( - 1 ) i n ! ( n - i ) ! c i + 1 x n - i = e c x i = 0 n ( - 1 ) n - i n ! i ! c n - i + 1 x i \int x^{n}e^{cx}\;\mathrm{d}x=\frac{1}{c}x^{n}e^{cx}-\frac{n}{c}\int x^{n-1}e^% {cx}\mathrm{d}x=\left(\frac{\partial}{\partial c}\right)^{n}\frac{e^{cx}}{c}=e% ^{cx}\sum_{i=0}^{n}(-1)^{i}\,\frac{n!}{(n-i)!\,c^{i+1}}\,x^{n-i}=e^{cx}\sum_{i% =0}^{n}(-1)^{n-i}\,\frac{n!}{i!\,c^{n-i+1}}\,x^{i}
  8. e c x x d x = ln | x | + n = 1 ( c x ) n n n ! \int\frac{e^{cx}}{x}\;\mathrm{d}x=\ln|x|+\sum_{n=1}^{\infty}\frac{(cx)^{n}}{n% \cdot n!}
  9. e c x x n d x = 1 n - 1 ( - e c x x n - 1 + c e c x x n - 1 d x ) (for n 1 ) \int\frac{e^{cx}}{x^{n}}\;\mathrm{d}x=\frac{1}{n-1}\left(-\frac{e^{cx}}{x^{n-1% }}+c\int\frac{e^{cx}}{x^{n-1}}\,\mathrm{d}x\right)\qquad\mbox{(for }~{}n\neq 1% \mbox{)}~{}
  10. e c x sin b x d x = e c x c 2 + b 2 ( c sin b x - b cos b x ) = e c x c 2 + b 2 sin ( b x - ϕ ) cos ( ϕ ) = c c 2 + b 2 \int e^{cx}\sin bx\;\mathrm{d}x=\frac{e^{cx}}{c^{2}+b^{2}}(c\sin bx-b\cos bx)=% \frac{e^{cx}}{\sqrt{c^{2}+b^{2}}}\sin(bx-\phi)\qquad\cos(\phi)=\frac{c}{\sqrt{% c^{2}+b^{2}}}
  11. e c x cos b x d x = e c x c 2 + b 2 ( c cos b x + b sin b x ) = e c x c 2 + b 2 cos ( b x - ϕ ) cos ( ϕ ) = c c 2 + b 2 \int e^{cx}\cos bx\;\mathrm{d}x=\frac{e^{cx}}{c^{2}+b^{2}}(c\cos bx+b\sin bx)=% \frac{e^{cx}}{\sqrt{c^{2}+b^{2}}}\cos(bx-\phi)\qquad\cos(\phi)=\frac{c}{\sqrt{% c^{2}+b^{2}}}
  12. e c x sin n x d x = e c x sin n - 1 x c 2 + n 2 ( c sin x - n cos x ) + n ( n - 1 ) c 2 + n 2 e c x sin n - 2 x d x \int e^{cx}\sin^{n}x\;\mathrm{d}x=\frac{e^{cx}\sin^{n-1}x}{c^{2}+n^{2}}(c\sin x% -n\cos x)+\frac{n(n-1)}{c^{2}+n^{2}}\int e^{cx}\sin^{n-2}x\;\mathrm{d}x
  13. e c x cos n x d x = e c x cos n - 1 x c 2 + n 2 ( c cos x + n sin x ) + n ( n - 1 ) c 2 + n 2 e c x cos n - 2 x d x \int e^{cx}\cos^{n}x\;\mathrm{d}x=\frac{e^{cx}\cos^{n-1}x}{c^{2}+n^{2}}(c\cos x% +n\sin x)+\frac{n(n-1)}{c^{2}+n^{2}}\int e^{cx}\cos^{n-2}x\;\mathrm{d}x
  14. e c x ln x d x = 1 c ( e c x ln | x | - Ei ( c x ) ) \int e^{cx}\ln x\;\mathrm{d}x=\frac{1}{c}\left(e^{cx}\ln|x|-\operatorname{Ei}% \,(cx)\right)
  15. x e c x 2 d x = 1 2 c e c x 2 \int xe^{cx^{2}}\;\mathrm{d}x=\frac{1}{2c}\;e^{cx^{2}}
  16. e - c x 2 d x = π 4 c erf ( c x ) \int e^{-cx^{2}}\;\mathrm{d}x=\sqrt{\frac{\pi}{4c}}\operatorname{erf}(\sqrt{c}x)
  17. erf \operatorname{erf}
  18. x e - c x 2 d x = - 1 2 c e - c x 2 \int xe^{-cx^{2}}\;\mathrm{d}x=-\frac{1}{2c}e^{-cx^{2}}
  19. e - x 2 x 2 d x = - e - x 2 x - π erf ( x ) \int\frac{e^{-x^{2}}}{x^{2}}\;\mathrm{d}x=-\frac{e^{-x^{2}}}{x}-\sqrt{\pi}% \mathrm{erf}(x)
  20. 1 σ 2 π e - 1 2 ( x - μ σ ) 2 d x = 1 2 ( erf x - μ σ 2 ) \int{\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}% \right)^{2}}}\;\mathrm{d}x=\frac{1}{2}\left(\operatorname{erf}\,\frac{x-\mu}{% \sigma\sqrt{2}}\right)
  21. e x 2 d x = e x 2 ( j = 0 n - 1 c 2 j 1 x 2 j + 1 ) + ( 2 n - 1 ) c 2 n - 2 e x 2 x 2 n d x valid for any n > 0 , \int e^{x^{2}}\,\mathrm{d}x=e^{x^{2}}\left(\sum_{j=0}^{n-1}c_{2j}\,\frac{1}{x^% {2j+1}}\right)+(2n-1)c_{2n-2}\int\frac{e^{x^{2}}}{x^{2n}}\;\mathrm{d}x\quad% \mbox{valid for any }~{}n>0,
  22. c 2 j = 1 3 5 ( 2 j - 1 ) 2 j + 1 = ( 2 j ) ! j ! 2 2 j + 1 . c_{2j}=\frac{1\cdot 3\cdot 5\cdots(2j-1)}{2^{j+1}}=\frac{(2j)\,!}{j!\,2^{2j+1}% }\ .
  23. n n
  24. x x x m d x = n = 0 m ( - 1 ) n ( n + 1 ) n - 1 n ! Γ ( n + 1 , - ln x ) + n = m + 1 ( - 1 ) n a m n Γ ( n + 1 , - ln x ) (for x > 0 ) {\int\underbrace{x^{x^{\cdot^{\cdot^{x}}}}}_{m}\,dx=\sum_{n=0}^{m}\frac{(-1)^{% n}(n+1)^{n-1}}{n!}\Gamma(n+1,-\ln x)+\sum_{n=m+1}^{\infty}(-1)^{n}a_{mn}\Gamma% (n+1,-\ln x)\qquad\mbox{(for }~{}x>0\mbox{)}~{}}
  25. a m n = { 1 if n = 0 , 1 n ! if m = 1 , 1 n j = 1 n j a m , n - j a m - 1 , j - 1 otherwise a_{mn}=\begin{cases}1&\,\text{if }n=0,\\ \frac{1}{n!}&\,\text{if }m=1,\\ \frac{1}{n}\sum_{j=1}^{n}ja_{m,n-j}a_{m-1,j-1}&\,\text{otherwise}\end{cases}
  26. Γ ( x , y ) \Gamma(x,y)
  27. 1 a e λ x + b d x = x b - 1 b λ ln ( a e λ x + b ) \int\frac{1}{ae^{\lambda x}+b}\;\mathrm{d}x=\frac{x}{b}-\frac{1}{b\lambda}\ln% \left(ae^{\lambda x}+b\right)\,
  28. b 0 b\neq 0
  29. λ 0 \lambda\neq 0
  30. a e λ x + b > 0 . ae^{\lambda x}+b>0\,.
  31. e 2 λ x a e λ x + b d x = 1 a 2 λ [ a e λ x + b - b ln ( a e λ x + b ) ] \int\frac{e^{2\lambda x}}{ae^{\lambda x}+b}\;\mathrm{d}x=\frac{1}{a^{2}\lambda% }\left[ae^{\lambda x}+b-b\ln\left(ae^{\lambda x}+b\right)\right]\,
  32. a 0 a\neq 0
  33. λ 0 \lambda\neq 0
  34. a e λ x + b > 0 . ae^{\lambda x}+b>0\,.
  35. 0 1 e x ln a + ( 1 - x ) ln b d x = 0 1 ( a b ) x b d x = 0 1 a x b 1 - x d x = a - b ln a - ln b \int_{0}^{1}e^{x\cdot\ln a+(1-x)\cdot\ln b}\;\mathrm{d}x=\int_{0}^{1}\left(% \frac{a}{b}\right)^{x}\cdot b\;\mathrm{d}x=\int_{0}^{1}a^{x}\cdot b^{1-x}\;% \mathrm{d}x=\frac{a-b}{\ln a-\ln b}
  36. a > 0 , b > 0 , a b a>0,\ b>0,\ a\neq b
  37. 0 e a x d x = 1 - a ( Re ( a ) < 0 ) \int_{0}^{\infty}e^{ax}\,\mathrm{d}x=\frac{1}{-a}\quad(\operatorname{Re}(a)<0)
  38. 0 e - a x 2 d x = 1 2 π a ( a > 0 ) \int_{0}^{\infty}e^{-ax^{2}}\,\mathrm{d}x=\frac{1}{2}\sqrt{\pi\over a}\quad(a>0)
  39. - e - a x 2 d x = π a ( a > 0 ) \int_{-\infty}^{\infty}e^{-ax^{2}}\,\mathrm{d}x=\sqrt{\pi\over a}\quad(a>0)
  40. - e - a x 2 e - 2 b x d x = π a e b 2 a ( a > 0 ) \int_{-\infty}^{\infty}e^{-ax^{2}}e^{-2bx}\,\mathrm{d}x=\sqrt{\frac{\pi}{a}}e^% {\frac{b^{2}}{a}}\quad(a>0)
  41. - x e - a ( x - b ) 2 d x = b π a ( Re ( a ) > 0 ) \int_{-\infty}^{\infty}xe^{-a(x-b)^{2}}\,\mathrm{d}x=b\sqrt{\frac{\pi}{a}}% \quad(\operatorname{Re}(a)>0)
  42. - x e - a x 2 + b x d x = π b 2 a 3 / 2 e b 2 4 a ( Re ( a ) > 0 ) \int_{-\infty}^{\infty}xe^{-ax^{2}+bx}\,\mathrm{d}x=\frac{\sqrt{\pi}b}{2a^{3/2% }}e^{\frac{b^{2}}{4a}}\quad(\operatorname{Re}(a)>0)
  43. - x 2 e - a x 2 d x = 1 2 π a 3 ( a > 0 ) \int_{-\infty}^{\infty}x^{2}e^{-ax^{2}}\,\mathrm{d}x=\frac{1}{2}\sqrt{\pi\over a% ^{3}}\quad(a>0)
  44. - x 2 e - a x 2 - b x d x = π ( 2 a + b 2 ) 4 a 5 / 2 e b 2 4 a ( Re ( a ) > 0 ) \int_{-\infty}^{\infty}x^{2}e^{-ax^{2}-bx}\,\mathrm{d}x=\frac{\sqrt{\pi}(2a+b^% {2})}{4a^{5/2}}e^{\frac{b^{2}}{4a}}\quad(\operatorname{Re}(a)>0)
  45. - x 3 e - a x 2 + b x d x = π ( 6 a + b 2 ) b 8 a 7 / 2 e b 2 4 a ( Re ( a ) > 0 ) \int_{-\infty}^{\infty}x^{3}e^{-ax^{2}+bx}\,\mathrm{d}x=\frac{\sqrt{\pi}(6a+b^% {2})b}{8a^{7/2}}e^{\frac{b^{2}}{4a}}\quad(\operatorname{Re}(a)>0)
  46. 0 x n e - a x 2 d x = { 1 2 Γ ( n + 1 2 ) / a n + 1 2 ( n > - 1 , a > 0 ) ( 2 k - 1 ) ! ! 2 k + 1 a k π a ( n = 2 k , k integer , a > 0 ) k ! 2 a k + 1 ( n = 2 k + 1 , k integer , a > 0 ) \int_{0}^{\infty}x^{n}e^{-ax^{2}}\,\mathrm{d}x=\begin{cases}\frac{1}{2}\Gamma% \left(\frac{n+1}{2}\right)/a^{\frac{n+1}{2}}&(n>-1,a>0)\\ \frac{(2k-1)!!}{2^{k+1}a^{k}}\sqrt{\frac{\pi}{a}}&(n=2k,k\;\,\text{integer},a>% 0)\\ \frac{k!}{2a^{k+1}}&(n=2k+1,k\;\,\text{integer},a>0)\end{cases}
  47. 0 x n e - a x d x = { Γ ( n + 1 ) a n + 1 ( n > - 1 , a > 0 ) n ! a n + 1 ( n = 0 , 1 , 2 , , a > 0 ) \int_{0}^{\infty}x^{n}e^{-ax}\,\mathrm{d}x=\begin{cases}\frac{\Gamma(n+1)}{a^{% n+1}}&(n>-1,a>0)\\ \frac{n!}{a^{n+1}}&(n=0,1,2,\ldots,a>0)\\ \end{cases}
  48. 0 1 x n e - a x d x = n ! a n + 1 [ 1 - e - a i = 0 n a i i ! ] \int_{0}^{1}x^{n}e^{-ax}\,\mathrm{d}x=\frac{n!}{a^{n+1}}\left[1-e^{-a}\sum_{i=% 0}^{n}\frac{a^{i}}{i!}\right]
  49. 0 e - a x b d x = 1 b a - 1 b Γ ( 1 b ) \int_{0}^{\infty}e^{-ax^{b}}dx=\frac{1}{b}\ a^{-\frac{1}{b}}\,\Gamma\left(% \frac{1}{b}\right)
  50. 0 x n e - a x b d x = 1 b a - n + 1 b Γ ( n + 1 b ) \int_{0}^{\infty}x^{n}e^{-ax^{b}}dx=\frac{1}{b}\ a^{-\frac{n+1}{b}}\,\Gamma% \left(\frac{n+1}{b}\right)
  51. 0 e - a x sin b x d x = b a 2 + b 2 ( a > 0 ) \int_{0}^{\infty}e^{-ax}\sin bx\,\mathrm{d}x=\frac{b}{a^{2}+b^{2}}\quad(a>0)
  52. 0 e - a x cos b x d x = a a 2 + b 2 ( a > 0 ) \int_{0}^{\infty}e^{-ax}\cos bx\,\mathrm{d}x=\frac{a}{a^{2}+b^{2}}\quad(a>0)
  53. 0 x e - a x sin b x d x = 2 a b ( a 2 + b 2 ) 2 ( a > 0 ) \int_{0}^{\infty}xe^{-ax}\sin bx\,\mathrm{d}x=\frac{2ab}{(a^{2}+b^{2})^{2}}% \quad(a>0)
  54. 0 x e - a x cos b x d x = a 2 - b 2 ( a 2 + b 2 ) 2 ( a > 0 ) \int_{0}^{\infty}xe^{-ax}\cos bx\,\mathrm{d}x=\frac{a^{2}-b^{2}}{(a^{2}+b^{2})% ^{2}}\quad(a>0)
  55. 0 2 π e x cos θ d θ = 2 π I 0 ( x ) \int_{0}^{2\pi}e^{x\cos\theta}d\theta=2\pi I_{0}(x)
  56. I 0 I_{0}
  57. 0 2 π e x cos θ + y sin θ d θ = 2 π I 0 ( x 2 + y 2 ) \int_{0}^{2\pi}e^{x\cos\theta+y\sin\theta}d\theta=2\pi I_{0}\left(\sqrt{x^{2}+% y^{2}}\right)

List_of_integrals_of_hyperbolic_functions.html

  1. sinh a x d x = 1 a cosh a x + C \int\sinh ax\,dx=\frac{1}{a}\cosh ax+C\,
  2. sinh 2 a x d x = 1 4 a sinh 2 a x - x 2 + C \int\sinh^{2}ax\,dx=\frac{1}{4a}\sinh 2ax-\frac{x}{2}+C\,
  3. sinh n a x d x = 1 a n sinh n - 1 a x cosh a x - n - 1 n sinh n - 2 a x d x (for n > 0 ) \int\sinh^{n}ax\,dx=\frac{1}{an}\sinh^{n-1}ax\cosh ax-\frac{n-1}{n}\int\sinh^{% n-2}ax\,dx\qquad\mbox{(for }~{}n>0\mbox{)}~{}\,
  4. sinh n a x d x = 1 a ( n + 1 ) sinh n + 1 a x cosh a x - n + 2 n + 1 sinh n + 2 a x d x (for n < 0 , n - 1 ) \int\sinh^{n}ax\,dx=\frac{1}{a(n+1)}\sinh^{n+1}ax\cosh ax-\frac{n+2}{n+1}\int% \sinh^{n+2}ax\,dx\qquad\mbox{(for }~{}n<0\mbox{, }~{}n\neq-1\mbox{)}~{}\,
  5. d x sinh a x = 1 a ln | tanh a x 2 | + C \int\frac{dx}{\sinh ax}=\frac{1}{a}\ln\left|\tanh\frac{ax}{2}\right|+C\,
  6. d x sinh a x = 1 a ln | cosh a x - 1 sinh a x | + C \int\frac{dx}{\sinh ax}=\frac{1}{a}\ln\left|\frac{\cosh ax-1}{\sinh ax}\right|% +C\,
  7. d x sinh a x = 1 a ln | sinh a x cosh a x + 1 | + C \int\frac{dx}{\sinh ax}=\frac{1}{a}\ln\left|\frac{\sinh ax}{\cosh ax+1}\right|% +C\,
  8. d x sinh a x = 1 2 a ln | cosh a x - 1 cosh a x + 1 | + C \int\frac{dx}{\sinh ax}=\frac{1}{2a}\ln\left|\frac{\cosh ax-1}{\cosh ax+1}% \right|+C\,
  9. d x sinh n a x = - cosh a x a ( n - 1 ) sinh n - 1 a x - n - 2 n - 1 d x sinh n - 2 a x (for n 1 ) \int\frac{dx}{\sinh^{n}ax}=-\frac{\cosh ax}{a(n-1)\sinh^{n-1}ax}-\frac{n-2}{n-% 1}\int\frac{dx}{\sinh^{n-2}ax}\qquad\mbox{(for }~{}n\neq 1\mbox{)}~{}\,
  10. x sinh a x d x = 1 a x cosh a x - 1 a 2 sinh a x + C \int x\sinh ax\,dx=\frac{1}{a}x\cosh ax-\frac{1}{a^{2}}\sinh ax+C\,
  11. sinh a x sinh b x d x = 1 a 2 - b 2 ( a sinh b x cosh a x - b cosh b x sinh a x ) + C (for a 2 b 2 ) \int\sinh ax\sinh bx\,dx=\frac{1}{a^{2}-b^{2}}(a\sinh bx\cosh ax-b\cosh bx% \sinh ax)+C\qquad\mbox{(for }~{}a^{2}\neq b^{2}\mbox{)}~{}\,
  12. cosh a x d x = 1 a sinh a x + C \int\cosh ax\,dx=\frac{1}{a}\sinh ax+C\,
  13. cosh 2 a x d x = 1 4 a sinh 2 a x + x 2 + C \int\cosh^{2}ax\,dx=\frac{1}{4a}\sinh 2ax+\frac{x}{2}+C\,
  14. cosh n a x d x = 1 a n sinh a x cosh n - 1 a x + n - 1 n cosh n - 2 a x d x (for n > 0 ) \int\cosh^{n}ax\,dx=\frac{1}{an}\sinh ax\cosh^{n-1}ax+\frac{n-1}{n}\int\cosh^{% n-2}ax\,dx\qquad\mbox{(for }~{}n>0\mbox{)}~{}\,
  15. cosh n a x d x = - 1 a ( n + 1 ) sinh a x cosh n + 1 a x + n + 2 n + 1 cosh n + 2 a x d x (for n < 0 , n - 1 ) \int\cosh^{n}ax\,dx=-\frac{1}{a(n+1)}\sinh ax\cosh^{n+1}ax+\frac{n+2}{n+1}\int% \cosh^{n+2}ax\,dx\qquad\mbox{(for }~{}n<0\mbox{, }~{}n\neq-1\mbox{)}~{}\,
  16. d x cosh a x = 2 a arctan e a x + C \int\frac{dx}{\cosh ax}=\frac{2}{a}\arctan e^{ax}+C\,
  17. d x cosh a x = 1 a arctan ( sinh a x ) + C \int\frac{dx}{\cosh ax}=\frac{1}{a}\arctan(\sinh ax)+C\,
  18. d x cosh n a x = sinh a x a ( n - 1 ) cosh n - 1 a x + n - 2 n - 1 d x cosh n - 2 a x (for n 1 ) \int\frac{dx}{\cosh^{n}ax}=\frac{\sinh ax}{a(n-1)\cosh^{n-1}ax}+\frac{n-2}{n-1% }\int\frac{dx}{\cosh^{n-2}ax}\qquad\mbox{(for }~{}n\neq 1\mbox{)}~{}\,
  19. x cosh a x d x = 1 a x sinh a x - 1 a 2 cosh a x + C \int x\cosh ax\,dx=\frac{1}{a}x\sinh ax-\frac{1}{a^{2}}\cosh ax+C\,
  20. x 2 cosh a x d x = - 2 x cosh a x a 2 + ( x 2 a + 2 a 3 ) sinh a x + C \int x^{2}\cosh ax\,dx=-\frac{2x\cosh ax}{a^{2}}+\left(\frac{x^{2}}{a}+\frac{2% }{a^{3}}\right)\sinh ax+C\,
  21. cosh a x cosh b x d x = 1 a 2 - b 2 ( a sinh a x cosh b x - b sinh b x cosh a x ) + C (for a 2 b 2 ) \int\cosh ax\cosh bx\,dx=\frac{1}{a^{2}-b^{2}}(a\sinh ax\cosh bx-b\sinh bx% \cosh ax)+C\qquad\mbox{(for }~{}a^{2}\neq b^{2}\mbox{)}~{}\,
  22. tanh x d x = ln cosh x + C \int\tanh x\,dx=\ln\cosh x+C
  23. tanh 2 a x d x = x - tanh a x a + C \int\tanh^{2}ax\,dx=x-\frac{\tanh ax}{a}+C\,
  24. tanh n a x d x = - 1 a ( n - 1 ) tanh n - 1 a x + tanh n - 2 a x d x (for n 1 ) \int\tanh^{n}ax\,dx=-\frac{1}{a(n-1)}\tanh^{n-1}ax+\int\tanh^{n-2}ax\,dx\qquad% \mbox{(for }~{}n\neq 1\mbox{)}~{}\,
  25. coth x d x = ln | sinh x | + C , for x 0 \int\coth x\,dx=\ln|\sinh x|+C,\,\text{ for }x\neq 0
  26. coth n a x d x = - 1 a ( n - 1 ) coth n - 1 a x + coth n - 2 a x d x (for n 1 ) \int\coth^{n}ax\,dx=-\frac{1}{a(n-1)}\coth^{n-1}ax+\int\coth^{n-2}ax\,dx\qquad% \mbox{(for }~{}n\neq 1\mbox{)}~{}\,
  27. sech x d x = arctan ( sinh x ) + C \int\operatorname{sech}\,x\,dx=\arctan\,(\sinh x)+C
  28. csch x d x = ln | tanh x 2 | + C , for x 0 \int\operatorname{csch}\,x\,dx=\ln\left|\tanh{x\over 2}\right|+C,\,\text{ for % }x\neq 0
  29. cosh a x sinh b x d x = 1 a 2 - b 2 ( a sinh a x sinh b x - b cosh a x cosh b x ) + C (for a 2 b 2 ) \int\cosh ax\sinh bx\,dx=\frac{1}{a^{2}-b^{2}}(a\sinh ax\sinh bx-b\cosh ax% \cosh bx)+C\qquad\mbox{(for }~{}a^{2}\neq b^{2}\mbox{)}~{}\,
  30. cosh n a x sinh m a x d x = cosh n - 1 a x a ( n - m ) sinh m - 1 a x + n - 1 n - m cosh n - 2 a x sinh m a x d x (for m n ) \int\frac{\cosh^{n}ax}{\sinh^{m}ax}dx=\frac{\cosh^{n-1}ax}{a(n-m)\sinh^{m-1}ax% }+\frac{n-1}{n-m}\int\frac{\cosh^{n-2}ax}{\sinh^{m}ax}dx\qquad\mbox{(for }~{}m% \neq n\mbox{)}~{}\,
  31. cosh n a x sinh m a x d x = - cosh n + 1 a x a ( m - 1 ) sinh m - 1 a x + n - m + 2 m - 1 cosh n a x sinh m - 2 a x d x (for m 1 ) \int\frac{\cosh^{n}ax}{\sinh^{m}ax}dx=-\frac{\cosh^{n+1}ax}{a(m-1)\sinh^{m-1}% ax}+\frac{n-m+2}{m-1}\int\frac{\cosh^{n}ax}{\sinh^{m-2}ax}dx\qquad\mbox{(for }% ~{}m\neq 1\mbox{)}~{}\,
  32. cosh n a x sinh m a x d x = - cosh n - 1 a x a ( m - 1 ) sinh m - 1 a x + n - 1 m - 1 cosh n - 2 a x sinh m - 2 a x d x (for m 1 ) \int\frac{\cosh^{n}ax}{\sinh^{m}ax}dx=-\frac{\cosh^{n-1}ax}{a(m-1)\sinh^{m-1}% ax}+\frac{n-1}{m-1}\int\frac{\cosh^{n-2}ax}{\sinh^{m-2}ax}dx\qquad\mbox{(for }% ~{}m\neq 1\mbox{)}~{}\,
  33. sinh m a x cosh n a x d x = sinh m - 1 a x a ( m - n ) cosh n - 1 a x + m - 1 n - m sinh m - 2 a x cosh n a x d x (for m n ) \int\frac{\sinh^{m}ax}{\cosh^{n}ax}dx=\frac{\sinh^{m-1}ax}{a(m-n)\cosh^{n-1}ax% }+\frac{m-1}{n-m}\int\frac{\sinh^{m-2}ax}{\cosh^{n}ax}dx\qquad\mbox{(for }~{}m% \neq n\mbox{)}~{}\,
  34. sinh m a x cosh n a x d x = sinh m + 1 a x a ( n - 1 ) cosh n - 1 a x + m - n + 2 n - 1 sinh m a x cosh n - 2 a x d x (for n 1 ) \int\frac{\sinh^{m}ax}{\cosh^{n}ax}dx=\frac{\sinh^{m+1}ax}{a(n-1)\cosh^{n-1}ax% }+\frac{m-n+2}{n-1}\int\frac{\sinh^{m}ax}{\cosh^{n-2}ax}dx\qquad\mbox{(for }~{% }n\neq 1\mbox{)}~{}\,
  35. sinh m a x cosh n a x d x = - sinh m - 1 a x a ( n - 1 ) cosh n - 1 a x + m - 1 n - 1 sinh m - 2 a x cosh n - 2 a x d x (for n 1 ) \int\frac{\sinh^{m}ax}{\cosh^{n}ax}dx=-\frac{\sinh^{m-1}ax}{a(n-1)\cosh^{n-1}% ax}+\frac{m-1}{n-1}\int\frac{\sinh^{m-2}ax}{\cosh^{n-2}ax}dx\qquad\mbox{(for }% ~{}n\neq 1\mbox{)}~{}\,
  36. sinh ( a x + b ) sin ( c x + d ) d x = a a 2 + c 2 cosh ( a x + b ) sin ( c x + d ) - c a 2 + c 2 sinh ( a x + b ) cos ( c x + d ) + C \int\sinh(ax+b)\sin(cx+d)\,dx=\frac{a}{a^{2}+c^{2}}\cosh(ax+b)\sin(cx+d)-\frac% {c}{a^{2}+c^{2}}\sinh(ax+b)\cos(cx+d)+C\,
  37. sinh ( a x + b ) cos ( c x + d ) d x = a a 2 + c 2 cosh ( a x + b ) cos ( c x + d ) + c a 2 + c 2 sinh ( a x + b ) sin ( c x + d ) + C \int\sinh(ax+b)\cos(cx+d)\,dx=\frac{a}{a^{2}+c^{2}}\cosh(ax+b)\cos(cx+d)+\frac% {c}{a^{2}+c^{2}}\sinh(ax+b)\sin(cx+d)+C\,
  38. cosh ( a x + b ) sin ( c x + d ) d x = a a 2 + c 2 sinh ( a x + b ) sin ( c x + d ) - c a 2 + c 2 cosh ( a x + b ) cos ( c x + d ) + C \int\cosh(ax+b)\sin(cx+d)\,dx=\frac{a}{a^{2}+c^{2}}\sinh(ax+b)\sin(cx+d)-\frac% {c}{a^{2}+c^{2}}\cosh(ax+b)\cos(cx+d)+C\,
  39. cosh ( a x + b ) cos ( c x + d ) d x = a a 2 + c 2 sinh ( a x + b ) cos ( c x + d ) + c a 2 + c 2 cosh ( a x + b ) sin ( c x + d ) + C \int\cosh(ax+b)\cos(cx+d)\,dx=\frac{a}{a^{2}+c^{2}}\sinh(ax+b)\cos(cx+d)+\frac% {c}{a^{2}+c^{2}}\cosh(ax+b)\sin(cx+d)+C\,

List_of_integrals_of_inverse_hyperbolic_functions.html

  1. arsinh ( a x ) d x = x arsinh ( a x ) - a 2 x 2 + 1 a + C \int\operatorname{arsinh}(a\,x)\,dx=x\,\operatorname{arsinh}(a\,x)-\frac{\sqrt% {a^{2}\,x^{2}+1}}{a}+C
  2. x arsinh ( a x ) d x = x 2 arsinh ( a x ) 2 + arsinh ( a x ) 4 a 2 - x a 2 x 2 + 1 4 a + C \int x\,\operatorname{arsinh}(a\,x)dx=\frac{x^{2}\,\operatorname{arsinh}(a\,x)% }{2}+\frac{\operatorname{arsinh}(a\,x)}{4\,a^{2}}-\frac{x\sqrt{a^{2}\,x^{2}+1}% }{4\,a}+C
  3. x 2 arsinh ( a x ) d x = x 3 arsinh ( a x ) 3 - ( a 2 x 2 - 2 ) a 2 x 2 + 1 9 a 3 + C \int x^{2}\,\operatorname{arsinh}(a\,x)dx=\frac{x^{3}\,\operatorname{arsinh}(a% \,x)}{3}-\frac{\left(a^{2}\,x^{2}-2\right)\sqrt{a^{2}\,x^{2}+1}}{9\,a^{3}}+C
  4. x m arsinh ( a x ) d x = x m + 1 arsinh ( a x ) m + 1 - a m + 1 x m + 1 a 2 x 2 + 1 d x ( m - 1 ) \int x^{m}\,\operatorname{arsinh}(a\,x)dx=\frac{x^{m+1}\,\operatorname{arsinh}% (a\,x)}{m+1}\,-\,\frac{a}{m+1}\int\frac{x^{m+1}}{\sqrt{a^{2}\,x^{2}+1}}\,dx% \quad(m\neq-1)
  5. arsinh ( a x ) 2 d x = 2 x + x arsinh ( a x ) 2 - 2 a 2 x 2 + 1 arsinh ( a x ) a + C \int\operatorname{arsinh}(a\,x)^{2}\,dx=2\,x+x\,\operatorname{arsinh}(a\,x)^{2% }-\frac{2\,\sqrt{a^{2}\,x^{2}+1}\,\operatorname{arsinh}(a\,x)}{a}+C
  6. arsinh ( a x ) n d x = x arsinh ( a x ) n - n a 2 x 2 + 1 arsinh ( a x ) n - 1 a + n ( n - 1 ) arsinh ( a x ) n - 2 d x \int\operatorname{arsinh}(a\,x)^{n}\,dx=x\,\operatorname{arsinh}(a\,x)^{n}\,-% \,\frac{n\,\sqrt{a^{2}\,x^{2}+1}\,\operatorname{arsinh}(a\,x)^{n-1}}{a}\,+\,n% \,(n-1)\int\operatorname{arsinh}(a\,x)^{n-2}\,dx
  7. arsinh ( a x ) n d x = - x arsinh ( a x ) n + 2 ( n + 1 ) ( n + 2 ) + a 2 x 2 + 1 arsinh ( a x ) n + 1 a ( n + 1 ) + 1 ( n + 1 ) ( n + 2 ) arsinh ( a x ) n + 2 d x ( n - 1 , - 2 ) \int\operatorname{arsinh}(a\,x)^{n}\,dx=-\frac{x\,\operatorname{arsinh}(a\,x)^% {n+2}}{(n+1)\,(n+2)}\,+\,\frac{\sqrt{a^{2}\,x^{2}+1}\,\operatorname{arsinh}(a% \,x)^{n+1}}{a(n+1)}\,+\,\frac{1}{(n+1)\,(n+2)}\int\operatorname{arsinh}(a\,x)^% {n+2}\,dx\quad(n\neq-1,-2)
  8. arcosh ( a x ) d x = x arcosh ( a x ) - a x + 1 a x - 1 a + C \int\operatorname{arcosh}(a\,x)\,dx=x\,\operatorname{arcosh}(a\,x)-\frac{\sqrt% {a\,x+1}\,\sqrt{a\,x-1}}{a}+C
  9. x arcosh ( a x ) d x = x 2 arcosh ( a x ) 2 - arcosh ( a x ) 4 a 2 - x a x + 1 a x - 1 4 a + C \int x\,\operatorname{arcosh}(a\,x)dx=\frac{x^{2}\,\operatorname{arcosh}(a\,x)% }{2}-\frac{\operatorname{arcosh}(a\,x)}{4\,a^{2}}-\frac{x\,\sqrt{a\,x+1}\,% \sqrt{a\,x-1}}{4\,a}+C
  10. x 2 arcosh ( a x ) d x = x 3 arcosh ( a x ) 3 - ( a 2 x 2 + 2 ) a x + 1 a x - 1 9 a 3 + C \int x^{2}\,\operatorname{arcosh}(a\,x)dx=\frac{x^{3}\,\operatorname{arcosh}(a% \,x)}{3}-\frac{\left(a^{2}\,x^{2}+2\right)\sqrt{a\,x+1}\,\sqrt{a\,x-1}}{9\,a^{% 3}}+C
  11. x m arcosh ( a x ) d x = x m + 1 arcosh ( a x ) m + 1 - a m + 1 x m + 1 a x + 1 a x - 1 d x ( m - 1 ) \int x^{m}\,\operatorname{arcosh}(a\,x)dx=\frac{x^{m+1}\,\operatorname{arcosh}% (a\,x)}{m+1}\,-\,\frac{a}{m+1}\int\frac{x^{m+1}}{\sqrt{a\,x+1}\,\sqrt{a\,x-1}}% \,dx\quad(m\neq-1)
  12. arcosh ( a x ) 2 d x = 2 x + x arcosh ( a x ) 2 - 2 a x + 1 a x - 1 arcosh ( a x ) a + C \int\operatorname{arcosh}(a\,x)^{2}\,dx=2\,x+x\,\operatorname{arcosh}(a\,x)^{2% }-\frac{2\,\sqrt{a\,x+1}\,\sqrt{a\,x-1}\,\operatorname{arcosh}(a\,x)}{a}+C
  13. arcosh ( a x ) n d x = x arcosh ( a x ) n - n a x + 1 a x - 1 arcosh ( a x ) n - 1 a + n ( n - 1 ) arcosh ( a x ) n - 2 d x \int\operatorname{arcosh}(a\,x)^{n}\,dx=x\,\operatorname{arcosh}(a\,x)^{n}\,-% \,\frac{n\,\sqrt{a\,x+1}\,\sqrt{a\,x-1}\,\operatorname{arcosh}(a\,x)^{n-1}}{a}% \,+\,n\,(n-1)\int\operatorname{arcosh}(a\,x)^{n-2}\,dx
  14. arcosh ( a x ) n d x = - x arcosh ( a x ) n + 2 ( n + 1 ) ( n + 2 ) + a x + 1 a x - 1 arcosh ( a x ) n + 1 a ( n + 1 ) + 1 ( n + 1 ) ( n + 2 ) arcosh ( a x ) n + 2 d x ( n - 1 , - 2 ) \int\operatorname{arcosh}(a\,x)^{n}\,dx=-\frac{x\,\operatorname{arcosh}(a\,x)^% {n+2}}{(n+1)\,(n+2)}\,+\,\frac{\sqrt{a\,x+1}\,\sqrt{a\,x-1}\,\operatorname{% arcosh}(a\,x)^{n+1}}{a\,(n+1)}\,+\,\frac{1}{(n+1)\,(n+2)}\int\operatorname{% arcosh}(a\,x)^{n+2}\,dx\quad(n\neq-1,-2)
  15. artanh ( a x ) d x = x artanh ( a x ) + ln ( 1 - a 2 x 2 ) 2 a + C \int\operatorname{artanh}(a\,x)\,dx=x\,\operatorname{artanh}(a\,x)+\frac{\ln% \left(1-a^{2}\,x^{2}\right)}{2\,a}+C
  16. x artanh ( a x ) d x = x 2 artanh ( a x ) 2 - artanh ( a x ) 2 a 2 + x 2 a + C \int x\,\operatorname{artanh}(a\,x)dx=\frac{x^{2}\,\operatorname{artanh}(a\,x)% }{2}-\frac{\operatorname{artanh}(a\,x)}{2\,a^{2}}+\frac{x}{2\,a}+C
  17. x 2 artanh ( a x ) d x = x 3 artanh ( a x ) 3 + ln ( 1 - a 2 x 2 ) 6 a 3 + x 2 6 a + C \int x^{2}\,\operatorname{artanh}(a\,x)dx=\frac{x^{3}\,\operatorname{artanh}(a% \,x)}{3}+\frac{\ln\left(1-a^{2}\,x^{2}\right)}{6\,a^{3}}+\frac{x^{2}}{6\,a}+C
  18. x m artanh ( a x ) d x = x m + 1 artanh ( a x ) m + 1 - a m + 1 x m + 1 1 - a 2 x 2 d x ( m - 1 ) \int x^{m}\,\operatorname{artanh}(a\,x)dx=\frac{x^{m+1}\operatorname{artanh}(a% \,x)}{m+1}-\frac{a}{m+1}\int\frac{x^{m+1}}{1-a^{2}\,x^{2}}\,dx\quad(m\neq-1)
  19. arcoth ( a x ) d x = x arcoth ( a x ) + ln ( a 2 x 2 - 1 ) 2 a + C \int\operatorname{arcoth}(a\,x)\,dx=x\,\operatorname{arcoth}(a\,x)+\frac{\ln% \left(a^{2}\,x^{2}-1\right)}{2\,a}+C
  20. x arcoth ( a x ) d x = x 2 arcoth ( a x ) 2 - arcoth ( a x ) 2 a 2 + x 2 a + C \int x\,\operatorname{arcoth}(a\,x)dx=\frac{x^{2}\,\operatorname{arcoth}(a\,x)% }{2}-\frac{\operatorname{arcoth}(a\,x)}{2\,a^{2}}+\frac{x}{2\,a}+C
  21. x 2 arcoth ( a x ) d x = x 3 arcoth ( a x ) 3 + ln ( a 2 x 2 - 1 ) 6 a 3 + x 2 6 a + C \int x^{2}\,\operatorname{arcoth}(a\,x)dx=\frac{x^{3}\,\operatorname{arcoth}(a% \,x)}{3}+\frac{\ln\left(a^{2}\,x^{2}-1\right)}{6\,a^{3}}+\frac{x^{2}}{6\,a}+C
  22. x m arcoth ( a x ) d x = x m + 1 arcoth ( a x ) m + 1 + a m + 1 x m + 1 a 2 x 2 - 1 d x ( m - 1 ) \int x^{m}\,\operatorname{arcoth}(a\,x)dx=\frac{x^{m+1}\operatorname{arcoth}(a% \,x)}{m+1}+\frac{a}{m+1}\int\frac{x^{m+1}}{a^{2}\,x^{2}-1}\,dx\quad(m\neq-1)
  23. arsech ( a x ) d x = x arsech ( a x ) - 2 a arctan 1 - a x 1 + a x + C \int\operatorname{arsech}(a\,x)\,dx=x\,\operatorname{arsech}(a\,x)-\frac{2}{a}% \,\operatorname{arctan}\sqrt{\frac{1-a\,x}{1+a\,x}}+C
  24. x arsech ( a x ) d x = x 2 arsech ( a x ) 2 - ( 1 + a x ) 2 a 2 1 - a x 1 + a x + C \int x\,\operatorname{arsech}(a\,x)dx=\frac{x^{2}\,\operatorname{arsech}(a\,x)% }{2}-\frac{(1+a\,x)}{2\,a^{2}}\sqrt{\frac{1-a\,x}{1+a\,x}}+C
  25. x 2 arsech ( a x ) d x = x 3 arsech ( a x ) 3 - 1 3 a 3 arctan 1 - a x 1 + a x - x ( 1 + a x ) 6 a 2 1 - a x 1 + a x + C \int x^{2}\,\operatorname{arsech}(a\,x)dx=\frac{x^{3}\,\operatorname{arsech}(a% \,x)}{3}\,-\,\frac{1}{3\,a^{3}}\,\operatorname{arctan}\sqrt{\frac{1-a\,x}{1+a% \,x}}\,-\,\frac{x(1+a\,x)}{6\,a^{2}}\sqrt{\frac{1-a\,x}{1+a\,x}}\,+\,C
  26. x m arsech ( a x ) d x = x m + 1 arsech ( a x ) m + 1 + 1 m + 1 x m ( 1 + a x ) 1 - a x 1 + a x d x ( m - 1 ) \int x^{m}\,\operatorname{arsech}(a\,x)dx=\frac{x^{m+1}\,\operatorname{arsech}% (a\,x)}{m+1}\,+\,\frac{1}{m+1}\int\frac{x^{m}}{(1+a\,x)\sqrt{\frac{1-a\,x}{1+a% \,x}}}\,dx\quad(m\neq-1)
  27. arcsch ( a x ) d x = x arcsch ( a x ) + 1 a arcoth 1 a 2 x 2 + 1 + C \int\operatorname{arcsch}(a\,x)\,dx=x\,\operatorname{arcsch}(a\,x)+\frac{1}{a}% \,\operatorname{arcoth}\sqrt{\frac{1}{a^{2}\,x^{2}}+1}+C
  28. x arcsch ( a x ) d x = x 2 arcsch ( a x ) 2 + x 2 a 1 a 2 x 2 + 1 + C \int x\,\operatorname{arcsch}(a\,x)dx=\frac{x^{2}\,\operatorname{arcsch}(a\,x)% }{2}+\frac{x}{2\,a}\sqrt{\frac{1}{a^{2}\,x^{2}}+1}+C
  29. x 2 arcsch ( a x ) d x = x 3 arcsch ( a x ) 3 - 1 6 a 3 arcoth 1 a 2 x 2 + 1 + x 2 6 a 1 a 2 x 2 + 1 + C \int x^{2}\,\operatorname{arcsch}(a\,x)dx=\frac{x^{3}\,\operatorname{arcsch}(a% \,x)}{3}\,-\,\frac{1}{6\,a^{3}}\,\operatorname{arcoth}\sqrt{\frac{1}{a^{2}\,x^% {2}}+1}\,+\,\frac{x^{2}}{6\,a}\sqrt{\frac{1}{a^{2}\,x^{2}}+1}\,+\,C
  30. x m arcsch ( a x ) d x = x m + 1 arcsch ( a x ) m + 1 + 1 a ( m + 1 ) x m - 1 1 a 2 x 2 + 1 d x ( m - 1 ) \int x^{m}\,\operatorname{arcsch}(a\,x)dx=\frac{x^{m+1}\operatorname{arcsch}(a% \,x)}{m+1}\,+\,\frac{1}{a(m+1)}\int\frac{x^{m-1}}{\sqrt{\frac{1}{a^{2}\,x^{2}}% +1}}\,dx\quad(m\neq-1)

List_of_integrals_of_inverse_trigonometric_functions.html

  1. arcsin ( x ) d x = x arcsin ( x ) + 1 - x 2 + C \int\arcsin(x)\,dx=x\arcsin(x)+{\sqrt{1-x^{2}}}+C
  2. arcsin ( a x ) d x = x arcsin ( a x ) + 1 - a 2 x 2 a + C \int\arcsin(a\,x)\,dx=x\arcsin(a\,x)+\frac{\sqrt{1-a^{2}\,x^{2}}}{a}+C
  3. x arcsin ( a x ) d x = x 2 arcsin ( a x ) 2 - arcsin ( a x ) 4 a 2 + x 1 - a 2 x 2 4 a + C \int x\arcsin(a\,x)\,dx=\frac{x^{2}\arcsin(a\,x)}{2}-\frac{\arcsin(a\,x)}{4\,a% ^{2}}+\frac{x\sqrt{1-a^{2}\,x^{2}}}{4\,a}+C
  4. x 2 arcsin ( a x ) d x = x 3 arcsin ( a x ) 3 + ( a 2 x 2 + 2 ) 1 - a 2 x 2 9 a 3 + C \int x^{2}\arcsin(a\,x)\,dx=\frac{x^{3}\arcsin(a\,x)}{3}+\frac{\left(a^{2}\,x^% {2}+2\right)\sqrt{1-a^{2}\,x^{2}}}{9\,a^{3}}+C
  5. x m arcsin ( a x ) d x = x m + 1 arcsin ( a x ) m + 1 - a m + 1 x m + 1 1 - a 2 x 2 d x ( m - 1 ) \int x^{m}\arcsin(a\,x)\,dx=\frac{x^{m+1}\arcsin(a\,x)}{m+1}\,-\,\frac{a}{m+1}% \int\frac{x^{m+1}}{\sqrt{1-a^{2}\,x^{2}}}\,dx\quad(m\neq-1)
  6. arcsin ( a x ) 2 d x = - 2 x + x arcsin ( a x ) 2 + 2 1 - a 2 x 2 arcsin ( a x ) a + C \int\arcsin(a\,x)^{2}\,dx=-2\,x+x\arcsin(a\,x)^{2}+\frac{2\sqrt{1-a^{2}\,x^{2}% }\arcsin(a\,x)}{a}+C
  7. arcsin ( a x ) n d x = x arcsin ( a x ) n + n 1 - a 2 x 2 arcsin ( a x ) n - 1 a - n ( n - 1 ) arcsin ( a x ) n - 2 d x \int\arcsin(a\,x)^{n}\,dx=x\arcsin(a\,x)^{n}\,+\,\frac{n\sqrt{1-a^{2}\,x^{2}}% \arcsin(a\,x)^{n-1}}{a}\,-\,n\,(n-1)\int\arcsin(a\,x)^{n-2}\,dx
  8. arcsin ( a x ) n d x = x arcsin ( a x ) n + 2 ( n + 1 ) ( n + 2 ) + 1 - a 2 x 2 arcsin ( a x ) n + 1 a ( n + 1 ) - 1 ( n + 1 ) ( n + 2 ) arcsin ( a x ) n + 2 d x ( n - 1 , - 2 ) \int\arcsin(a\,x)^{n}\,dx=\frac{x\arcsin(a\,x)^{n+2}}{(n+1)\,(n+2)}\,+\,\frac{% \sqrt{1-a^{2}\,x^{2}}\arcsin(a\,x)^{n+1}}{a\,(n+1)}\,-\,\frac{1}{(n+1)\,(n+2)}% \int\arcsin(a\,x)^{n+2}\,dx\quad(n\neq-1,-2)
  9. arccos ( x ) d x = x arccos ( x ) - 1 - x 2 + C \int\arccos(x)\,dx=x\arccos(x)-{\sqrt{1-x^{2}}}+C
  10. arccos ( a x ) d x = x arccos ( a x ) - 1 - a 2 x 2 a + C \int\arccos(a\,x)\,dx=x\arccos(a\,x)-\frac{\sqrt{1-a^{2}\,x^{2}}}{a}+C
  11. x arccos ( a x ) d x = x 2 arccos ( a x ) 2 - arccos ( a x ) 4 a 2 - x 1 - a 2 x 2 4 a + C \int x\arccos(a\,x)\,dx=\frac{x^{2}\arccos(a\,x)}{2}-\frac{\arccos(a\,x)}{4\,a% ^{2}}-\frac{x\sqrt{1-a^{2}\,x^{2}}}{4\,a}+C
  12. x 2 arccos ( a x ) d x = x 3 arccos ( a x ) 3 - ( a 2 x 2 + 2 ) 1 - a 2 x 2 9 a 3 + C \int x^{2}\arccos(a\,x)\,dx=\frac{x^{3}\arccos(a\,x)}{3}-\frac{\left(a^{2}\,x^% {2}+2\right)\sqrt{1-a^{2}\,x^{2}}}{9\,a^{3}}+C
  13. x m arccos ( a x ) d x = x m + 1 arccos ( a x ) m + 1 + a m + 1 x m + 1 1 - a 2 x 2 d x ( m - 1 ) \int x^{m}\arccos(a\,x)\,dx=\frac{x^{m+1}\arccos(a\,x)}{m+1}\,+\,\frac{a}{m+1}% \int\frac{x^{m+1}}{\sqrt{1-a^{2}\,x^{2}}}\,dx\quad(m\neq-1)
  14. arccos ( a x ) 2 d x = - 2 x + x arccos ( a x ) 2 - 2 1 - a 2 x 2 arccos ( a x ) a + C \int\arccos(a\,x)^{2}\,dx=-2\,x+x\arccos(a\,x)^{2}-\frac{2\sqrt{1-a^{2}\,x^{2}% }\arccos(a\,x)}{a}+C
  15. arccos ( a x ) n d x = x arccos ( a x ) n - n 1 - a 2 x 2 arccos ( a x ) n - 1 a - n ( n - 1 ) arccos ( a x ) n - 2 d x \int\arccos(a\,x)^{n}\,dx=x\arccos(a\,x)^{n}\,-\,\frac{n\sqrt{1-a^{2}\,x^{2}}% \arccos(a\,x)^{n-1}}{a}\,-\,n\,(n-1)\int\arccos(a\,x)^{n-2}\,dx
  16. arccos ( a x ) n d x = x arccos ( a x ) n + 2 ( n + 1 ) ( n + 2 ) - 1 - a 2 x 2 arccos ( a x ) n + 1 a ( n + 1 ) - 1 ( n + 1 ) ( n + 2 ) arccos ( a x ) n + 2 d x ( n - 1 , - 2 ) \int\arccos(a\,x)^{n}\,dx=\frac{x\arccos(a\,x)^{n+2}}{(n+1)\,(n+2)}\,-\,\frac{% \sqrt{1-a^{2}\,x^{2}}\arccos(a\,x)^{n+1}}{a\,(n+1)}\,-\,\frac{1}{(n+1)\,(n+2)}% \int\arccos(a\,x)^{n+2}\,dx\quad(n\neq-1,-2)
  17. arctan ( x ) d x = x arctan ( x ) - ln ( x 2 + 1 ) 2 + C \int\arctan(x)\,dx=x\arctan(x)-\frac{\ln\left(x^{2}+1\right)}{2}+C
  18. arctan ( a x ) d x = x arctan ( a x ) - ln ( a 2 x 2 + 1 ) 2 a + C \int\arctan(a\,x)\,dx=x\arctan(a\,x)-\frac{\ln\left(a^{2}\,x^{2}+1\right)}{2\,% a}+C
  19. x arctan ( a x ) d x = x 2 arctan ( a x ) 2 + arctan ( a x ) 2 a 2 - x 2 a + C \int x\arctan(a\,x)\,dx=\frac{x^{2}\arctan(a\,x)}{2}+\frac{\arctan(a\,x)}{2\,a% ^{2}}-\frac{x}{2\,a}+C
  20. x 2 arctan ( a x ) d x = x 3 arctan ( a x ) 3 + ln ( a 2 x 2 + 1 ) 6 a 3 - x 2 6 a + C \int x^{2}\arctan(a\,x)\,dx=\frac{x^{3}\arctan(a\,x)}{3}+\frac{\ln\left(a^{2}% \,x^{2}+1\right)}{6\,a^{3}}-\frac{x^{2}}{6\,a}+C
  21. x m arctan ( a x ) d x = x m + 1 arctan ( a x ) m + 1 - a m + 1 x m + 1 a 2 x 2 + 1 d x ( m - 1 ) \int x^{m}\arctan(a\,x)\,dx=\frac{x^{m+1}\arctan(a\,x)}{m+1}-\frac{a}{m+1}\int% \frac{x^{m+1}}{a^{2}\,x^{2}+1}\,dx\quad(m\neq-1)
  22. \arccot ( x ) d x = x \arccot ( x ) + ln ( x 2 + 1 ) 2 + C \int\arccot(x)\,dx=x\arccot(x)+\frac{\ln\left(x^{2}+1\right)}{2}+C
  23. \arccot ( a x ) d x = x \arccot ( a x ) + ln ( a 2 x 2 + 1 ) 2 a + C \int\arccot(a\,x)\,dx=x\arccot(a\,x)+\frac{\ln\left(a^{2}\,x^{2}+1\right)}{2\,% a}+C
  24. x \arccot ( a x ) d x = x 2 \arccot ( a x ) 2 + \arccot ( a x ) 2 a 2 + x 2 a + C \int x\arccot(a\,x)\,dx=\frac{x^{2}\arccot(a\,x)}{2}+\frac{\arccot(a\,x)}{2\,a% ^{2}}+\frac{x}{2\,a}+C
  25. x 2 \arccot ( a x ) d x = x 3 \arccot ( a x ) 3 - ln ( a 2 x 2 + 1 ) 6 a 3 + x 2 6 a + C \int x^{2}\arccot(a\,x)\,dx=\frac{x^{3}\arccot(a\,x)}{3}-\frac{\ln\left(a^{2}% \,x^{2}+1\right)}{6\,a^{3}}+\frac{x^{2}}{6\,a}+C
  26. x m \arccot ( a x ) d x = x m + 1 \arccot ( a x ) m + 1 + a m + 1 x m + 1 a 2 x 2 + 1 d x ( m - 1 ) \int x^{m}\arccot(a\,x)\,dx=\frac{x^{m+1}\arccot(a\,x)}{m+1}+\frac{a}{m+1}\int% \frac{x^{m+1}}{a^{2}\,x^{2}+1}\,dx\quad(m\neq-1)
  27. \arcsec ( x ) d x = x \arcsec ( x ) - artanh 1 - 1 x 2 + C \int\arcsec(x)\,dx=x\arcsec(x)-\operatorname{artanh}\,\sqrt{1-\frac{1}{x^{2}}}+C
  28. \arcsec ( a x ) d x = x \arcsec ( a x ) - 1 a artanh 1 - 1 a 2 x 2 + C \int\arcsec(a\,x)\,dx=x\arcsec(a\,x)-\frac{1}{a}\,\operatorname{artanh}\,\sqrt% {1-\frac{1}{a^{2}\,x^{2}}}+C
  29. x \arcsec ( a x ) d x = x 2 \arcsec ( a x ) 2 - x 2 a 1 - 1 a 2 x 2 + C \int x\arcsec(a\,x)\,dx=\frac{x^{2}\arcsec(a\,x)}{2}-\frac{x}{2\,a}\sqrt{1-% \frac{1}{a^{2}\,x^{2}}}+C
  30. x 2 \arcsec ( a x ) d x = x 3 \arcsec ( a x ) 3 - 1 6 a 3 artanh 1 - 1 a 2 x 2 - x 2 6 a 1 - 1 a 2 x 2 + C \int x^{2}\arcsec(a\,x)\,dx=\frac{x^{3}\arcsec(a\,x)}{3}\,-\,\frac{1}{6\,a^{3}% }\,\operatorname{artanh}\,\sqrt{1-\frac{1}{a^{2}\,x^{2}}}\,-\,\frac{x^{2}}{6\,% a}\sqrt{1-\frac{1}{a^{2}\,x^{2}}}\,+\,C
  31. x m \arcsec ( a x ) d x = x m + 1 \arcsec ( a x ) m + 1 - 1 a ( m + 1 ) x m - 1 1 - 1 a 2 x 2 d x ( m - 1 ) \int x^{m}\arcsec(a\,x)\,dx=\frac{x^{m+1}\arcsec(a\,x)}{m+1}\,-\,\frac{1}{a\,(% m+1)}\int\frac{x^{m-1}}{\sqrt{1-\frac{1}{a^{2}\,x^{2}}}}\,dx\quad(m\neq-1)
  32. \arccsc ( x ) d x = x \arccsc ( x ) + ln | x + x 2 - 1 | + C = x \arccsc ( x ) + arcosh ( x ) + C \int\arccsc(x)\,dx=x\arccsc(x)\,+\,\ln\left|x+\sqrt{x^{2}-1}\right|\,+\,C=x% \arccsc(x)\,+\,\operatorname{arcosh}(x)\,+\,C
  33. \arccsc ( a x ) d x = x \arccsc ( a x ) + 1 a artanh 1 - 1 a 2 x 2 + C \int\arccsc(a\,x)\,dx=x\arccsc(a\,x)+\frac{1}{a}\,\operatorname{artanh}\,\sqrt% {1-\frac{1}{a^{2}\,x^{2}}}+C
  34. x \arccsc ( a x ) d x = x 2 \arccsc ( a x ) 2 + x 2 a 1 - 1 a 2 x 2 + C \int x\arccsc(a\,x)\,dx=\frac{x^{2}\arccsc(a\,x)}{2}+\frac{x}{2\,a}\sqrt{1-% \frac{1}{a^{2}\,x^{2}}}+C
  35. x 2 \arccsc ( a x ) d x = x 3 \arccsc ( a x ) 3 + 1 6 a 3 artanh 1 - 1 a 2 x 2 + x 2 6 a 1 - 1 a 2 x 2 + C \int x^{2}\arccsc(a\,x)\,dx=\frac{x^{3}\arccsc(a\,x)}{3}\,+\,\frac{1}{6\,a^{3}% }\,\operatorname{artanh}\,\sqrt{1-\frac{1}{a^{2}\,x^{2}}}\,+\,\frac{x^{2}}{6\,% a}\sqrt{1-\frac{1}{a^{2}\,x^{2}}}\,+\,C
  36. x m \arccsc ( a x ) d x = x m + 1 \arccsc ( a x ) m + 1 + 1 a ( m + 1 ) x m - 1 1 - 1 a 2 x 2 d x ( m - 1 ) \int x^{m}\arccsc(a\,x)\,dx=\frac{x^{m+1}\arccsc(a\,x)}{m+1}\,+\,\frac{1}{a\,(% m+1)}\int\frac{x^{m-1}}{\sqrt{1-\frac{1}{a^{2}\,x^{2}}}}\,dx\quad(m\neq-1)

List_of_integrals_of_irrational_functions.html

  1. r d x = 1 2 ( x r + a 2 ln ( x + r ) ) \int r\;dx=\frac{1}{2}\left(xr+a^{2}\,\ln\left(x+r\right)\right)
  2. r 3 d x = 1 4 x r 3 + 3 8 a 2 x r + 3 8 a 4 ln ( x + r ) \int r^{3}\;dx=\frac{1}{4}xr^{3}+\frac{3}{8}a^{2}xr+\frac{3}{8}a^{4}\ln\left(x% +r\right)
  3. r 5 d x = 1 6 x r 5 + 5 24 a 2 x r 3 + 5 16 a 4 x r + 5 16 a 6 ln ( x + r ) \int r^{5}\;dx=\frac{1}{6}xr^{5}+\frac{5}{24}a^{2}xr^{3}+\frac{5}{16}a^{4}xr+% \frac{5}{16}a^{6}\ln\left(x+r\right)
  4. x r d x = r 3 3 \int xr\;dx=\frac{r^{3}}{3}
  5. x r 3 d x = r 5 5 \int xr^{3}\;dx=\frac{r^{5}}{5}
  6. x r 2 n + 1 d x = r 2 n + 3 2 n + 3 \int xr^{2n+1}\;dx=\frac{r^{2n+3}}{2n+3}
  7. x 2 r d x = x r 3 4 - a 2 x r 8 - a 4 8 ln ( x + r ) \int x^{2}r\;dx=\frac{xr^{3}}{4}-\frac{a^{2}xr}{8}-\frac{a^{4}}{8}\ln\left(x+r\right)
  8. x 2 r 3 d x = x r 5 6 - a 2 x r 3 24 - a 4 x r 16 - a 6 16 ln ( x + r ) \int x^{2}r^{3}\;dx=\frac{xr^{5}}{6}-\frac{a^{2}xr^{3}}{24}-\frac{a^{4}xr}{16}% -\frac{a^{6}}{16}\ln\left(x+r\right)
  9. x 3 r d x = r 5 5 - a 2 r 3 3 \int x^{3}r\;dx=\frac{r^{5}}{5}-\frac{a^{2}r^{3}}{3}
  10. x 3 r 3 d x = r 7 7 - a 2 r 5 5 \int x^{3}r^{3}\;dx=\frac{r^{7}}{7}-\frac{a^{2}r^{5}}{5}
  11. x 3 r 2 n + 1 d x = r 2 n + 5 2 n + 5 - a 2 r 2 n + 3 2 n + 3 \int x^{3}r^{2n+1}\;dx=\frac{r^{2n+5}}{2n+5}-\frac{a^{2}r^{2n+3}}{2n+3}
  12. x 4 r d x = x 3 r 3 6 - a 2 x r 3 8 + a 4 x r 16 + a 6 16 ln ( x + r ) \int x^{4}r\;dx=\frac{x^{3}r^{3}}{6}-\frac{a^{2}xr^{3}}{8}+\frac{a^{4}xr}{16}+% \frac{a^{6}}{16}\ln\left(x+r\right)
  13. x 4 r 3 d x = x 3 r 5 8 - a 2 x r 5 16 + a 4 x r 3 64 + 3 a 6 x r 128 + 3 a 8 128 ln ( x + r ) \int x^{4}r^{3}\;dx=\frac{x^{3}r^{5}}{8}-\frac{a^{2}xr^{5}}{16}+\frac{a^{4}xr^% {3}}{64}+\frac{3a^{6}xr}{128}+\frac{3a^{8}}{128}\ln\left(x+r\right)
  14. x 5 r d x = r 7 7 - 2 a 2 r 5 5 + a 4 r 3 3 \int x^{5}r\;dx=\frac{r^{7}}{7}-\frac{2a^{2}r^{5}}{5}+\frac{a^{4}r^{3}}{3}
  15. x 5 r 3 d x = r 9 9 - 2 a 2 r 7 7 + a 4 r 5 5 \int x^{5}r^{3}\;dx=\frac{r^{9}}{9}-\frac{2a^{2}r^{7}}{7}+\frac{a^{4}r^{5}}{5}
  16. x 5 r 2 n + 1 d x = r 2 n + 7 2 n + 7 - 2 a 2 r 2 n + 5 2 n + 5 + a 4 r 2 n + 3 2 n + 3 \int x^{5}r^{2n+1}\;dx=\frac{r^{2n+7}}{2n+7}-\frac{2a^{2}r^{2n+5}}{2n+5}+\frac% {a^{4}r^{2n+3}}{2n+3}
  17. r d x x = r - a ln | a + r x | = r - a arsinh a x \int\frac{r\;dx}{x}=r-a\ln\left|\frac{a+r}{x}\right|=r-a\,\operatorname{arsinh% }\frac{a}{x}
  18. r 3 d x x = r 3 3 + a 2 r - a 3 ln | a + r x | \int\frac{r^{3}\;dx}{x}=\frac{r^{3}}{3}+a^{2}r-a^{3}\ln\left|\frac{a+r}{x}\right|
  19. r 5 d x x = r 5 5 + a 2 r 3 3 + a 4 r - a 5 ln | a + r x | \int\frac{r^{5}\;dx}{x}=\frac{r^{5}}{5}+\frac{a^{2}r^{3}}{3}+a^{4}r-a^{5}\ln% \left|\frac{a+r}{x}\right|
  20. r 7 d x x = r 7 7 + a 2 r 5 5 + a 4 r 3 3 + a 6 r - a 7 ln | a + r x | \int\frac{r^{7}\;dx}{x}=\frac{r^{7}}{7}+\frac{a^{2}r^{5}}{5}+\frac{a^{4}r^{3}}% {3}+a^{6}r-a^{7}\ln\left|\frac{a+r}{x}\right|
  21. d x r = arsinh x a = ln ( x + r a ) \int\frac{dx}{r}=\operatorname{arsinh}\frac{x}{a}=\ln\left(\frac{x+r}{a}\right)
  22. d x r 3 = x a 2 r \int\frac{dx}{r^{3}}=\frac{x}{a^{2}r}
  23. x d x r = r \int\frac{x\,dx}{r}=r
  24. x d x r 3 = - 1 r \int\frac{x\,dx}{r^{3}}=-\frac{1}{r}
  25. x 2 d x r = x 2 r - a 2 2 arsinh x a = x 2 r - a 2 2 ln ( x + r a ) \int\frac{x^{2}\;dx}{r}=\frac{x}{2}r-\frac{a^{2}}{2}\,\operatorname{arsinh}% \frac{x}{a}=\frac{x}{2}r-\frac{a^{2}}{2}\ln\left(\frac{x+r}{a}\right)
  26. d x x r = - 1 a arsinh a x = - 1 a ln | a + r x | \int\frac{dx}{xr}=-\frac{1}{a}\,\operatorname{arsinh}\frac{a}{x}=-\frac{1}{a}% \ln\left|\frac{a+r}{x}\right|
  27. ( x 2 > a 2 ) (x^{2}>a^{2})
  28. ( x 2 < a 2 ) (x^{2}<a^{2})
  29. s d x = 1 2 ( x s - a 2 ln ( x + s ) ) \int s\;dx=\frac{1}{2}\left(xs-a^{2}\ln(x+s)\right)
  30. x s d x = 1 3 s 3 \int xs\;dx=\frac{1}{3}s^{3}
  31. s d x x = s - a arccos | a x | \int\frac{s\;dx}{x}=s-a\arccos\left|\frac{a}{x}\right|
  32. d x s = ln | x + s a | \int\frac{dx}{s}=\ln\left|\frac{x+s}{a}\right|
  33. ln | x + s a | = sgn ( x ) arcosh | x a | = 1 2 ln ( x + s x - s ) \ln\left|\frac{x+s}{a}\right|=\mathrm{sgn}(x)\,\operatorname{arcosh}\left|% \frac{x}{a}\right|=\frac{1}{2}\ln\left(\frac{x+s}{x-s}\right)
  34. arcosh | x a | \operatorname{arcosh}\left|\frac{x}{a}\right|
  35. x d x s = s \int\frac{x\;dx}{s}=s
  36. x d x s 3 = - 1 s \int\frac{x\;dx}{s^{3}}=-\frac{1}{s}
  37. x d x s 5 = - 1 3 s 3 \int\frac{x\;dx}{s^{5}}=-\frac{1}{3s^{3}}
  38. x d x s 7 = - 1 5 s 5 \int\frac{x\;dx}{s^{7}}=-\frac{1}{5s^{5}}
  39. x d x s 2 n + 1 = - 1 ( 2 n - 1 ) s 2 n - 1 \int\frac{x\;dx}{s^{2n+1}}=-\frac{1}{(2n-1)s^{2n-1}}
  40. x 2 m d x s 2 n + 1 = - 1 2 n - 1 x 2 m - 1 s 2 n - 1 + 2 m - 1 2 n - 1 x 2 m - 2 d x s 2 n - 1 \int\frac{x^{2m}\;dx}{s^{2n+1}}=-\frac{1}{2n-1}\frac{x^{2m-1}}{s^{2n-1}}+\frac% {2m-1}{2n-1}\int\frac{x^{2m-2}\;dx}{s^{2n-1}}
  41. x 2 d x s = x s 2 + a 2 2 ln | x + s a | \int\frac{x^{2}\;dx}{s}=\frac{xs}{2}+\frac{a^{2}}{2}\ln\left|\frac{x+s}{a}\right|
  42. x 2 d x s 3 = - x s + ln | x + s a | \int\frac{x^{2}\;dx}{s^{3}}=-\frac{x}{s}+\ln\left|\frac{x+s}{a}\right|
  43. x 4 d x s = x 3 s 4 + 3 8 a 2 x s + 3 8 a 4 ln | x + s a | \int\frac{x^{4}\;dx}{s}=\frac{x^{3}s}{4}+\frac{3}{8}a^{2}xs+\frac{3}{8}a^{4}% \ln\left|\frac{x+s}{a}\right|
  44. x 4 d x s 3 = x s 2 - a 2 x s + 3 2 a 2 ln | x + s a | \int\frac{x^{4}\;dx}{s^{3}}=\frac{xs}{2}-\frac{a^{2}x}{s}+\frac{3}{2}a^{2}\ln% \left|\frac{x+s}{a}\right|
  45. x 4 d x s 5 = - x s - 1 3 x 3 s 3 + ln | x + s a | \int\frac{x^{4}\;dx}{s^{5}}=-\frac{x}{s}-\frac{1}{3}\frac{x^{3}}{s^{3}}+\ln% \left|\frac{x+s}{a}\right|
  46. x 2 m d x s 2 n + 1 = ( - 1 ) n - m 1 a 2 ( n - m ) i = 0 n - m - 1 1 2 ( m + i ) + 1 ( n - m - 1 i ) x 2 ( m + i ) + 1 s 2 ( m + i ) + 1 ( n > m 0 ) \int\frac{x^{2m}\;dx}{s^{2n+1}}=(-1)^{n-m}\frac{1}{a^{2(n-m)}}\sum_{i=0}^{n-m-% 1}\frac{1}{2(m+i)+1}{n-m-1\choose i}\frac{x^{2(m+i)+1}}{s^{2(m+i)+1}}\qquad% \mbox{(}~{}n>m\geq 0\mbox{)}~{}
  47. d x s 3 = - 1 a 2 x s \int\frac{dx}{s^{3}}=-\frac{1}{a^{2}}\frac{x}{s}
  48. d x s 5 = 1 a 4 [ x s - 1 3 x 3 s 3 ] \int\frac{dx}{s^{5}}=\frac{1}{a^{4}}\left[\frac{x}{s}-\frac{1}{3}\frac{x^{3}}{% s^{3}}\right]
  49. d x s 7 = - 1 a 6 [ x s - 2 3 x 3 s 3 + 1 5 x 5 s 5 ] \int\frac{dx}{s^{7}}=-\frac{1}{a^{6}}\left[\frac{x}{s}-\frac{2}{3}\frac{x^{3}}% {s^{3}}+\frac{1}{5}\frac{x^{5}}{s^{5}}\right]
  50. d x s 9 = 1 a 8 [ x s - 3 3 x 3 s 3 + 3 5 x 5 s 5 - 1 7 x 7 s 7 ] \int\frac{dx}{s^{9}}=\frac{1}{a^{8}}\left[\frac{x}{s}-\frac{3}{3}\frac{x^{3}}{% s^{3}}+\frac{3}{5}\frac{x^{5}}{s^{5}}-\frac{1}{7}\frac{x^{7}}{s^{7}}\right]
  51. x 2 d x s 5 = - 1 a 2 x 3 3 s 3 \int\frac{x^{2}\;dx}{s^{5}}=-\frac{1}{a^{2}}\frac{x^{3}}{3s^{3}}
  52. x 2 d x s 7 = 1 a 4 [ 1 3 x 3 s 3 - 1 5 x 5 s 5 ] \int\frac{x^{2}\;dx}{s^{7}}=\frac{1}{a^{4}}\left[\frac{1}{3}\frac{x^{3}}{s^{3}% }-\frac{1}{5}\frac{x^{5}}{s^{5}}\right]
  53. x 2 d x s 9 = - 1 a 6 [ 1 3 x 3 s 3 - 2 5 x 5 s 5 + 1 7 x 7 s 7 ] \int\frac{x^{2}\;dx}{s^{9}}=-\frac{1}{a^{6}}\left[\frac{1}{3}\frac{x^{3}}{s^{3% }}-\frac{2}{5}\frac{x^{5}}{s^{5}}+\frac{1}{7}\frac{x^{7}}{s^{7}}\right]
  54. u d x = 1 2 ( x u + a 2 arcsin x a ) ( | x | | a | ) \int u\;dx=\frac{1}{2}\left(xu+a^{2}\arcsin\frac{x}{a}\right)\qquad\mbox{(}~{}% |x|\leq|a|\mbox{)}~{}
  55. x u d x = - 1 3 u 3 ( | x | | a | ) \int xu\;dx=-\frac{1}{3}u^{3}\qquad\mbox{(}~{}|x|\leq|a|\mbox{)}~{}
  56. x 2 u d x = - x 4 u 3 + a 2 8 ( x u + a 2 arcsin x a ) ( | x | | a | ) \int x^{2}u\;dx=-\frac{x}{4}u^{3}+\frac{a^{2}}{8}(xu+a^{2}\arcsin\frac{x}{a})% \qquad\mbox{(}~{}|x|\leq|a|\mbox{)}~{}
  57. u d x x = u - a ln | a + u x | ( | x | | a | ) \int\frac{u\;dx}{x}=u-a\ln\left|\frac{a+u}{x}\right|\qquad\mbox{(}~{}|x|\leq|a% |\mbox{)}~{}
  58. d x u = arcsin x a ( | x | | a | ) \int\frac{dx}{u}=\arcsin\frac{x}{a}\qquad\mbox{(}~{}|x|\leq|a|\mbox{)}~{}
  59. x 2 d x u = 1 2 ( - x u + a 2 arcsin x a ) ( | x | | a | ) \int\frac{x^{2}\;dx}{u}=\frac{1}{2}\left(-xu+a^{2}\arcsin\frac{x}{a}\right)% \qquad\mbox{(}~{}|x|\leq|a|\mbox{)}~{}
  60. u d x = 1 2 ( x u - sgn x arcosh | x a | ) (for | x | | a | ) \int u\;dx=\frac{1}{2}\left(xu-\operatorname{sgn}x\,\operatorname{arcosh}\left% |\frac{x}{a}\right|\right)\qquad\mbox{(for }~{}|x|\geq|a|\mbox{)}~{}
  61. x u d x = - u ( | x | | a | ) \int\frac{x}{u}\;dx=-u\qquad\mbox{(}~{}|x|\leq|a|\mbox{)}~{}
  62. d x R = 1 a ln | 2 a R + 2 a x + b | (for a > 0 ) \int\frac{dx}{R}=\frac{1}{\sqrt{a}}\ln\left|2\sqrt{a}R+2ax+b\right|\qquad\mbox% {(for }~{}a>0\mbox{)}~{}
  63. d x R = 1 a arsinh 2 a x + b 4 a c - b 2 (for a > 0 , 4 a c - b 2 > 0 ) \int\frac{dx}{R}=\frac{1}{\sqrt{a}}\,\operatorname{arsinh}\frac{2ax+b}{\sqrt{4% ac-b^{2}}}\qquad\mbox{(for }~{}a>0\mbox{, }~{}4ac-b^{2}>0\mbox{)}~{}
  64. d x R = 1 a ln | 2 a x + b | (for a > 0 , 4 a c - b 2 = 0 ) \int\frac{dx}{R}=\frac{1}{\sqrt{a}}\ln|2ax+b|\quad\mbox{(for }~{}a>0\mbox{, }~% {}4ac-b^{2}=0\mbox{)}~{}
  65. d x R = - 1 - a arcsin 2 a x + b b 2 - 4 a c (for a < 0 , 4 a c - b 2 < 0 , | 2 a x + b | < b 2 - 4 a c ) \int\frac{dx}{R}=-\frac{1}{\sqrt{-a}}\arcsin\frac{2ax+b}{\sqrt{b^{2}-4ac}}% \qquad\mbox{(for }~{}a<0\mbox{, }~{}4ac-b^{2}<0\mbox{, }~{}\left|2ax+b\right|<% \sqrt{b^{2}-4ac}\mbox{)}~{}
  66. d x R 3 = 4 a x + 2 b ( 4 a c - b 2 ) R \int\frac{dx}{R^{3}}=\frac{4ax+2b}{(4ac-b^{2})R}
  67. d x R 5 = 4 a x + 2 b 3 ( 4 a c - b 2 ) R ( 1 R 2 + 8 a 4 a c - b 2 ) \int\frac{dx}{R^{5}}=\frac{4ax+2b}{3(4ac-b^{2})R}\left(\frac{1}{R^{2}}+\frac{8% a}{4ac-b^{2}}\right)
  68. d x R 2 n + 1 = 2 ( 2 n - 1 ) ( 4 a c - b 2 ) ( 2 a x + b R 2 n - 1 + 4 a ( n - 1 ) d x R 2 n - 1 ) \int\frac{dx}{R^{2n+1}}=\frac{2}{(2n-1)(4ac-b^{2})}\left(\frac{2ax+b}{R^{2n-1}% }+4a(n-1)\int\frac{dx}{R^{2n-1}}\right)
  69. x R d x = R a - b 2 a d x R \int\frac{x}{R}\;dx=\frac{R}{a}-\frac{b}{2a}\int\frac{dx}{R}
  70. x R 3 d x = - 2 b x + 4 c ( 4 a c - b 2 ) R \int\frac{x}{R^{3}}\;dx=-\frac{2bx+4c}{(4ac-b^{2})R}
  71. x R 2 n + 1 d x = - 1 ( 2 n - 1 ) a R 2 n - 1 - b 2 a d x R 2 n + 1 \int\frac{x}{R^{2n+1}}\;dx=-\frac{1}{(2n-1)aR^{2n-1}}-\frac{b}{2a}\int\frac{dx% }{R^{2n+1}}
  72. d x x R = - 1 c ln | 2 c R + b x + 2 c x | , c > 0 \int\frac{dx}{xR}=-\frac{1}{\sqrt{c}}\ln\left|\frac{2\sqrt{c}R+bx+2c}{x}\right% |,~{}c>0
  73. d x x R = - 1 c arsinh ( b x + 2 c | x | 4 a c - b 2 ) , c < 0 \int\frac{dx}{xR}=-\frac{1}{\sqrt{c}}\operatorname{arsinh}\left(\frac{bx+2c}{|% x|\sqrt{4ac-b^{2}}}\right),~{}c<0
  74. d x x R = 1 - c arcsin ( b x + 2 c | x | b 2 - 4 a c ) , c < 0 , b 2 - 4 a c > 0 \int\frac{dx}{xR}=\frac{1}{\sqrt{-c}}\operatorname{arcsin}\left(\frac{bx+2c}{|% x|\sqrt{b^{2}-4ac}}\right),~{}c<0,b^{2}-4ac>0
  75. d x x R = - 2 b x ( a x 2 + b x ) , c = 0 \int\frac{dx}{xR}=-\frac{2}{bx}\left(\sqrt{ax^{2}+bx}\right),~{}c=0
  76. v 1 u 2 - b v\equiv\frac{1}{\sqrt{u^{2}-b}}
  77. - 2 a v d v 1 + b v 2 -2\sqrt{a}\int\frac{vdv}{\sqrt{1+bv^{2}}}
  78. x d x r = r \int\frac{x\;dx}{r}=r
  79. u a x + b a x u\equiv\sqrt{\frac{ax+b}{ax}}
  80. d x x R = - 2 a b d u = - 2 a b u \int\frac{dx}{xR}=-\frac{2\sqrt{a}}{b}\int du=-\frac{2\sqrt{a}}{b}u
  81. x 2 R d x = 2 a x - 3 b 4 a 2 R + 3 b 2 - 4 a c 8 a 2 d x R \int\frac{x^{2}}{R}\;dx=\frac{2ax-3b}{4a^{2}}R+\frac{3b^{2}-4ac}{8a^{2}}\int% \frac{dx}{R}
  82. d x x 2 R = - R c x - b 2 c d x x R \int\frac{dx}{x^{2}R}=-\frac{R}{cx}-\frac{b}{2c}\int\frac{dx}{xR}
  83. R d x = 2 a x + b 4 a R + 4 a c - b 2 8 a d x R \int R\,dx=\frac{2ax+b}{4a}R+\frac{4ac-b^{2}}{8a}\int\frac{dx}{R}
  84. x R d x = R 3 3 a - b ( 2 a x + b ) 8 a 2 R - b ( 4 a c - b 2 ) 16 a 2 d x R \int xR\,dx=\frac{R^{3}}{3a}-\frac{b(2ax+b)}{8a^{2}}R-\frac{b(4ac-b^{2})}{16a^% {2}}\int\frac{dx}{R}
  85. x 2 R d x = 6 a x - 5 b 24 a 2 R 3 + 5 b 2 - 4 a c 16 a 2 R d x \int x^{2}R\,dx=\frac{6ax-5b}{24a^{2}}R^{3}+\frac{5b^{2}-4ac}{16a^{2}}\int R\,dx
  86. R x d x = R + b 2 d x R + c d x x R \int\frac{R}{x}\,dx=R+\frac{b}{2}\int\frac{dx}{R}+c\int\frac{dx}{xR}
  87. R x 2 d x = - R x + a d x R 2 + b 2 d x R \int\frac{R}{x^{2}}\,dx=-\frac{R}{x}+a\int\frac{dx}{R^{2}}+\frac{b}{2}\int% \frac{dx}{R}
  88. x 2 d x R 3 = ( 2 b 2 - 4 a c ) x + 2 b c a ( 4 a c - b 2 ) R + 1 a d x R \int\frac{x^{2}\,dx}{R^{3}}=\frac{(2b^{2}-4ac)x+2bc}{a(4ac-b^{2})R}+\frac{1}{a% }\int\frac{dx}{R}
  89. S d x = 2 S 3 3 a \int S{dx}=\frac{2S^{3}}{3a}
  90. d x S = 2 S a \int\frac{dx}{S}=\frac{2S}{a}
  91. d x x S = { - 2 b arcoth ( S b ) (for b > 0 , a x > 0 ) - 2 b artanh ( S b ) (for b > 0 , a x < 0 ) 2 - b arctan ( S - b ) (for b < 0 ) \int\frac{dx}{xS}=\begin{cases}-\frac{2}{\sqrt{b}}\mathrm{arcoth}\left(\frac{S% }{\sqrt{b}}\right)&\mbox{(for }~{}b>0,\quad ax>0\mbox{)}\\ -\frac{2}{\sqrt{b}}\mathrm{artanh}\left(\frac{S}{\sqrt{b}}\right)&\mbox{(for }% ~{}b>0,\quad ax<0\mbox{)}\\ \frac{2}{\sqrt{-b}}\arctan\left(\frac{S}{\sqrt{-b}}\right)&\mbox{(for }~{}b<0% \mbox{)}\\ \end{cases}
  92. S x d x = { 2 ( S - b arcoth ( S b ) ) (for b > 0 , a x > 0 ) 2 ( S - b artanh ( S b ) ) (for b > 0 , a x < 0 ) 2 ( S - - b arctan ( S - b ) ) (for b < 0 ) \int\frac{S}{x}\,dx=\begin{cases}2\left(S-\sqrt{b}\,\mathrm{arcoth}\left(\frac% {S}{\sqrt{b}}\right)\right)&\mbox{(for }~{}b>0,\quad ax>0\mbox{)}\\ 2\left(S-\sqrt{b}\,\mathrm{artanh}\left(\frac{S}{\sqrt{b}}\right)\right)&\mbox% {(for }~{}b>0,\quad ax<0\mbox{)}\\ 2\left(S-\sqrt{-b}\arctan\left(\frac{S}{\sqrt{-b}}\right)\right)&\mbox{(for }~% {}b<0\mbox{)}\\ \end{cases}
  93. x n S d x = 2 a ( 2 n + 1 ) ( x n S - b n x n - 1 S d x ) \int\frac{x^{n}}{S}dx=\frac{2}{a(2n+1)}\left(x^{n}S-bn\int\frac{x^{n-1}}{S}dx\right)
  94. x n S d x = 2 a ( 2 n + 3 ) ( x n S 3 - n b x n - 1 S d x ) \int x^{n}Sdx=\frac{2}{a(2n+3)}\left(x^{n}S^{3}-nb\int x^{n-1}Sdx\right)
  95. 1 x n S d x = - 1 b ( n - 1 ) ( S x n - 1 + ( n - 3 2 ) a d x x n - 1 S ) \int\frac{1}{x^{n}S}dx=-\frac{1}{b(n-1)}\left(\frac{S}{x^{n-1}}+\left(n-\frac{% 3}{2}\right)a\int\frac{dx}{x^{n-1}S}\right)

List_of_integrals_of_logarithmic_functions.html

  1. ln a x d x = x ln a x - x \int\ln ax\;dx=x\ln ax-x
  2. ln ( a x + b ) d x = ( a x + b ) ln ( a x + b ) - a x a \int\ln(ax+b)\;dx=\frac{(ax+b)\ln(ax+b)-ax}{a}
  3. ( ln x ) 2 d x = x ( ln x ) 2 - 2 x ln x + 2 x \int(\ln x)^{2}\;dx=x(\ln x)^{2}-2x\ln x+2x
  4. ( ln x ) n d x = x k = 0 n ( - 1 ) n - k n ! k ! ( ln x ) k = Θ ( x ( ln x ) n ) \int(\ln x)^{n}\;dx=x\sum^{n}_{k=0}(-1)^{n-k}\frac{n!}{k!}(\ln x)^{k}=\Theta(x% (\ln x)^{n})
  5. d x ln x = ln | ln x | + ln x + k = 2 ( ln x ) k k k ! \int\frac{dx}{\ln x}=\ln|\ln x|+\ln x+\sum^{\infty}_{k=2}\frac{(\ln x)^{k}}{k% \cdot k!}
  6. d x ln x = li ( x ) \int\frac{dx}{\ln x}=\,\text{li}(x)
  7. li ( x ) = Θ ( x ln x ) \,\text{li}(x)=\Theta(\frac{x}{\ln x})
  8. d x ( ln x ) n = - x ( n - 1 ) ( ln x ) n - 1 + 1 n - 1 d x ( ln x ) n - 1 (for n 1 ) \int\frac{dx}{(\ln x)^{n}}=-\frac{x}{(n-1)(\ln x)^{n-1}}+\frac{1}{n-1}\int% \frac{dx}{(\ln x)^{n-1}}\qquad\mbox{(for }~{}n\neq 1\mbox{)}~{}
  9. x m ln x d x = x m + 1 ( ln x m + 1 - 1 ( m + 1 ) 2 ) (for m - 1 ) \int x^{m}\ln x\;dx=x^{m+1}\left(\frac{\ln x}{m+1}-\frac{1}{(m+1)^{2}}\right)% \qquad\mbox{(for }~{}m\neq-1\mbox{)}~{}
  10. x m ( ln x ) n d x = x m + 1 ( ln x ) n m + 1 - n m + 1 x m ( ln x ) n - 1 d x (for m - 1 ) \int x^{m}(\ln x)^{n}\;dx=\frac{x^{m+1}(\ln x)^{n}}{m+1}-\frac{n}{m+1}\int x^{% m}(\ln x)^{n-1}dx\qquad\mbox{(for }~{}m\neq-1\mbox{)}~{}
  11. ( ln x ) n d x x = ( ln x ) n + 1 n + 1 (for n - 1 ) \int\frac{(\ln x)^{n}\;dx}{x}=\frac{(\ln x)^{n+1}}{n+1}\qquad\mbox{(for }~{}n% \neq-1\mbox{)}~{}
  12. ln x n d x x = ( ln x n ) 2 2 n (for n 0 ) \int\frac{\ln{x^{n}}\;dx}{x}=\frac{(\ln{x^{n}})^{2}}{2n}\qquad\mbox{(for }~{}n% \neq 0\mbox{)}~{}
  13. ln x d x x m = - ln x ( m - 1 ) x m - 1 - 1 ( m - 1 ) 2 x m - 1 (for m 1 ) \int\frac{\ln x\,dx}{x^{m}}=-\frac{\ln x}{(m-1)x^{m-1}}-\frac{1}{(m-1)^{2}x^{m% -1}}\qquad\mbox{(for }~{}m\neq 1\mbox{)}~{}
  14. ( ln x ) n d x x m = - ( ln x ) n ( m - 1 ) x m - 1 + n m - 1 ( ln x ) n - 1 d x x m (for m 1 ) \int\frac{(\ln x)^{n}\;dx}{x^{m}}=-\frac{(\ln x)^{n}}{(m-1)x^{m-1}}+\frac{n}{m% -1}\int\frac{(\ln x)^{n-1}dx}{x^{m}}\qquad\mbox{(for }~{}m\neq 1\mbox{)}~{}
  15. x m d x ( ln x ) n = - x m + 1 ( n - 1 ) ( ln x ) n - 1 + m + 1 n - 1 x m d x ( ln x ) n - 1 (for n 1 ) \int\frac{x^{m}\;dx}{(\ln x)^{n}}=-\frac{x^{m+1}}{(n-1)(\ln x)^{n-1}}+\frac{m+% 1}{n-1}\int\frac{x^{m}dx}{(\ln x)^{n-1}}\qquad\mbox{(for }~{}n\neq 1\mbox{)}~{}
  16. d x x ln x = ln | ln x | \int\frac{dx}{x\ln x}=\ln\left|\ln x\right|
  17. d x x ln x ln ln x = ln | ln | ln x | | \int\frac{dx}{x\ln x\ln\ln x}=\ln\left|\ln\left|\ln x\right|\right|
  18. d x x ln ln x = li ( ln x ) \int\frac{dx}{x\ln\ln x}=\,\text{li}(\ln x)
  19. d x x n ln x = ln | ln x | + k = 1 ( - 1 ) k ( n - 1 ) k ( ln x ) k k k ! \int\frac{dx}{x^{n}\ln x}=\ln\left|\ln x\right|+\sum^{\infty}_{k=1}(-1)^{k}% \frac{(n-1)^{k}(\ln x)^{k}}{k\cdot k!}
  20. d x x ( ln x ) n = - 1 ( n - 1 ) ( ln x ) n - 1 (for n 1 ) \int\frac{dx}{x(\ln x)^{n}}=-\frac{1}{(n-1)(\ln x)^{n-1}}\qquad\mbox{(for }~{}% n\neq 1\mbox{)}~{}
  21. ln ( x 2 + a 2 ) d x = x ln ( x 2 + a 2 ) - 2 x + 2 a tan - 1 x a \int\ln(x^{2}+a^{2})\;dx=x\ln(x^{2}+a^{2})-2x+2a\tan^{-1}\frac{x}{a}
  22. x x 2 + a 2 ln ( x 2 + a 2 ) d x = 1 4 ln 2 ( x 2 + a 2 ) \int\frac{x}{x^{2}+a^{2}}\ln(x^{2}+a^{2})\;dx=\frac{1}{4}\ln^{2}(x^{2}+a^{2})
  23. sin ( ln x ) d x = x 2 ( sin ( ln x ) - cos ( ln x ) ) \int\sin(\ln x)\;dx=\frac{x}{2}(\sin(\ln x)-\cos(\ln x))
  24. cos ( ln x ) d x = x 2 ( sin ( ln x ) + cos ( ln x ) ) \int\cos(\ln x)\;dx=\frac{x}{2}(\sin(\ln x)+\cos(\ln x))
  25. e x ( x ln x - x - 1 x ) d x = e x ( x ln x - x - ln x ) \int e^{x}\left(x\ln x-x-\frac{1}{x}\right)\;dx=e^{x}(x\ln x-x-\ln x)
  26. 1 e x ( 1 x - ln x ) d x = ln x e x \int\frac{1}{e^{x}}\left(\frac{1}{x}-\ln x\right)\;dx=\frac{\ln x}{e^{x}}
  27. e x ( 1 ln x - 1 x ln 2 x ) d x = e x ln x \int e^{x}\left(\frac{1}{\ln x}-\frac{1}{x\ln^{2}x}\right)\;dx=\frac{e^{x}}{% \ln x}
  28. n n
  29. ln x d x = x ( ln x - 1 ) + C 0 \int\ln x\;dx=x\;(\ln x-1)+C_{0}
  30. ln x d x d x = x n n ! ( ln x - k = 1 n 1 k ) + k = 0 n - 1 C k x k k ! \int\cdot\cdot\cdot\int\ln x\;dx\cdot\cdot\cdot\;dx=\frac{x^{n}}{n!}\left(\ln% \,x-\sum_{k=1}^{n}\frac{1}{k}\right)+\sum_{k=0}^{n-1}C_{k}\frac{x^{k}}{k!}

List_of_integrals_of_rational_functions.html

  1. f ( x ) f ( x ) d x = ln | f ( x ) | + C \int\frac{f^{\prime}(x)}{f(x)}\,dx=\ln\left|f(x)\right|+C
  2. 1 x 2 + a 2 d x = 1 a arctan x a + C \int\frac{1}{x^{2}+a^{2}}\,dx=\frac{1}{a}\arctan\frac{x}{a}\,\!+C
  3. 1 x 2 - a 2 d x = { - 1 a artanh x a = 1 2 a ln a - x a + x + C (for | x | < | a | ) - 1 a arcoth x a = 1 2 a ln x - a x + a + C (for | x | > | a | ) \int\frac{1}{x^{2}-a^{2}}\,dx=\begin{cases}\displaystyle-\frac{1}{a}\,% \operatorname{artanh}\frac{x}{a}=\frac{1}{2a}\ln\frac{a-x}{a+x}+C&\,\text{(for% }|x|<|a|\mbox{)}\\ \displaystyle-\frac{1}{a}\,\operatorname{arcoth}\frac{x}{a}=\frac{1}{2a}\ln% \frac{x-a}{x+a}+C&\,\text{(for }|x|>|a|\mbox{)}\end{cases}
  4. d x x 2 n + 1 = k = 1 2 n - 1 { 1 2 n - 1 [ sin ( ( 2 k - 1 ) π 2 n ) arctan [ ( x - cos ( ( 2 k - 1 ) π 2 n ) ) csc ( ( 2 k - 1 ) π 2 n ) ] ] - 1 2 n [ cos ( ( 2 k - 1 ) π 2 n ) ln | x 2 - 2 x cos ( ( 2 k - 1 ) π 2 n ) + 1 | ] } + C \int\frac{dx}{x^{2^{n}}+1}=\sum_{k=1}^{2^{n-1}}\left\{\frac{1}{2^{n-1}}\left[% \sin\left(\frac{(2k-1)\pi}{2^{n}}\right)\arctan\left[\left(x-\cos\left(\frac{(% 2k-1)\pi}{2^{n}}\right)\right)\csc\left(\frac{(2k-1)\pi}{2^{n}}\right)\right]% \right]-\frac{1}{2^{n}}\left[\cos\left(\frac{(2k-1)\pi}{2^{n}}\right)\ln\left|% x^{2}-2x\cos\left(\frac{(2k-1)\pi}{2^{n}}\right)+1\right|\right]\right\}+C
  5. a ( x - b ) n \frac{a}{(x-b)^{n}}
  6. a x + b ( ( x - c ) 2 + d 2 ) n . \frac{ax+b}{\left((x-c)^{2}+d^{2}\right)^{n}}.
  7. 1 a x + b d x = 1 a ln | a x + b | + C \int\frac{1}{ax+b}\,dx=\frac{1}{a}\ln\left|ax+b\right|+C
  8. 1 a x + b d x = { 1 a ln | a x + b | + C - x < - b / a 1 a ln | a x + b | + C + x > - b / a \int\frac{1}{ax+b}\,dx=\begin{cases}\frac{1}{a}\ln\left|ax+b\right|+C^{-}&x<-b% /a\\ \frac{1}{a}\ln\left|ax+b\right|+C^{+}&x>-b/a\end{cases}
  9. ( a x + b ) n d x = ( a x + b ) n + 1 a ( n + 1 ) + C (for n - 1 ) \int(ax+b)^{n}\,dx=\frac{(ax+b)^{n+1}}{a(n+1)}+C\qquad\,\text{(for }n\neq-1% \mbox{)}~{}\,\!
  10. x a x + b d x = x a - b a 2 ln | a x + b | + C \int\frac{x}{ax+b}\,dx=\frac{x}{a}-\frac{b}{a^{2}}\ln\left|ax+b\right|+C
  11. x ( a x + b ) 2 d x = b a 2 ( a x + b ) + 1 a 2 ln | a x + b | + C \int\frac{x}{(ax+b)^{2}}\,dx=\frac{b}{a^{2}(ax+b)}+\frac{1}{a^{2}}\ln\left|ax+% b\right|+C
  12. x ( a x + b ) n d x = a ( 1 - n ) x - b a 2 ( n - 1 ) ( n - 2 ) ( a x + b ) n - 1 + C (for n { 1 , 2 } ) \int\frac{x}{(ax+b)^{n}}\,dx=\frac{a(1-n)x-b}{a^{2}(n-1)(n-2)(ax+b)^{n-1}}+C% \qquad\,\text{(for }n\not\in\{1,2\}\mbox{)}~{}
  13. x ( a x + b ) n d x = a ( n + 1 ) x - b a 2 ( n + 1 ) ( n + 2 ) ( a x + b ) n + 1 + C (for n { - 1 , - 2 } ) \int x(ax+b)^{n}\,dx=\frac{a(n+1)x-b}{a^{2}(n+1)(n+2)}(ax+b)^{n+1}+C\qquad\,% \text{(for }n\not\in\{-1,-2\}\mbox{)}~{}
  14. x 2 a x + b d x = b 2 ln ( | a x + b | ) a 3 + a x 2 - 2 b x 2 a 2 + C \int\frac{x^{2}}{ax+b}\,dx=\frac{b^{2}\ln(\left|ax+b\right|)}{a^{3}}+\frac{ax^% {2}-2bx}{2a^{2}}+C
  15. x 2 ( a x + b ) 2 d x = 1 a 3 ( a x - 2 b ln | a x + b | - b 2 a x + b ) + C \int\frac{x^{2}}{(ax+b)^{2}}\,dx=\frac{1}{a^{3}}\left(ax-2b\ln\left|ax+b\right% |-\frac{b^{2}}{ax+b}\right)+C
  16. x 2 ( a x + b ) 3 d x = 1 a 3 ( ln | a x + b | + 2 b a x + b - b 2 2 ( a x + b ) 2 ) + C \int\frac{x^{2}}{(ax+b)^{3}}\,dx=\frac{1}{a^{3}}\left(\ln\left|ax+b\right|+% \frac{2b}{ax+b}-\frac{b^{2}}{2(ax+b)^{2}}\right)+C
  17. x 2 ( a x + b ) n d x = 1 a 3 ( - ( a x + b ) 3 - n ( n - 3 ) + 2 b ( a x + b ) 2 - n ( n - 2 ) - b 2 ( a x + b ) 1 - n ( n - 1 ) ) + C (for n { 1 , 2 , 3 } ) \int\frac{x^{2}}{(ax+b)^{n}}\,dx=\frac{1}{a^{3}}\left(-\frac{(ax+b)^{3-n}}{(n-% 3)}+\frac{2b(ax+b)^{2-n}}{(n-2)}-\frac{b^{2}(ax+b)^{1-n}}{(n-1)}\right)+C% \qquad\,\text{(for }n\not\in\{1,2,3\}\mbox{)}~{}
  18. 1 x ( a x + b ) d x = - 1 b ln | a x + b x | + C \int\frac{1}{x(ax+b)}\,dx=-\frac{1}{b}\ln\left|\frac{ax+b}{x}\right|+C
  19. 1 x 2 ( a x + b ) d x = - 1 b x + a b 2 ln | a x + b x | + C \int\frac{1}{x^{2}(ax+b)}\,dx=-\frac{1}{bx}+\frac{a}{b^{2}}\ln\left|\frac{ax+b% }{x}\right|+C
  20. 1 x 2 ( a x + b ) 2 d x = - a ( 1 b 2 ( a x + b ) + 1 a b 2 x - 2 b 3 ln | a x + b x | ) + C \int\frac{1}{x^{2}(ax+b)^{2}}\,dx=-a\left(\frac{1}{b^{2}(ax+b)}+\frac{1}{ab^{2% }x}-\frac{2}{b^{3}}\ln\left|\frac{ax+b}{x}\right|\right)+C
  21. a 0 : a\neq 0:
  22. 1 a x 2 + b x + c d x = { 2 4 a c - b 2 arctan 2 a x + b 4 a c - b 2 + C (for 4 a c - b 2 > 0 ) - 2 b 2 - 4 a c arctanh 2 a x + b b 2 - 4 a c + C = 1 b 2 - 4 a c ln | 2 a x + b - b 2 - 4 a c 2 a x + b + b 2 - 4 a c | + C (for 4 a c - b 2 < 0 ) - 2 2 a x + b + C (for 4 a c - b 2 = 0 ) \int\frac{1}{ax^{2}+bx+c}dx=\begin{cases}\displaystyle\frac{2}{\sqrt{4ac-b^{2}% }}\arctan\frac{2ax+b}{\sqrt{4ac-b^{2}}}+C&\,\text{(for }4ac-b^{2}>0\mbox{)}\\ \displaystyle-\frac{2}{\sqrt{b^{2}-4ac}}\,\mathrm{arctanh}\frac{2ax+b}{\sqrt{b% ^{2}-4ac}}+C=\frac{1}{\sqrt{b^{2}-4ac}}\ln\left|\frac{2ax+b-\sqrt{b^{2}-4ac}}{% 2ax+b+\sqrt{b^{2}-4ac}}\right|+C&\,\text{(for }4ac-b^{2}<0\mbox{)}\\ \displaystyle-\frac{2}{2ax+b}+C&\,\text{(for }4ac-b^{2}=0\mbox{)}\end{cases}
  23. x a x 2 + b x + c d x = 1 2 a ln | a x 2 + b x + c | - b 2 a d x a x 2 + b x + c + C \int\frac{x}{ax^{2}+bx+c}\,dx=\frac{1}{2a}\ln\left|ax^{2}+bx+c\right|-\frac{b}% {2a}\int\frac{dx}{ax^{2}+bx+c}+C
  24. m x + n a x 2 + b x + c d x = { m 2 a ln | a x 2 + b x + c | + 2 a n - b m a 4 a c - b 2 arctan 2 a x + b 4 a c - b 2 + C (for 4 a c - b 2 > 0 ) m 2 a ln | a x 2 + b x + c | - 2 a n - b m a b 2 - 4 a c arctanh 2 a x + b b 2 - 4 a c + C (for 4 a c - b 2 < 0 ) m 2 a ln | a x 2 + b x + c | - 2 a n - b m a ( 2 a x + b ) + C (for 4 a c - b 2 = 0 ) \int\frac{mx+n}{ax^{2}+bx+c}\,dx=\begin{cases}\displaystyle\frac{m}{2a}\ln% \left|ax^{2}+bx+c\right|+\frac{2an-bm}{a\sqrt{4ac-b^{2}}}\arctan\frac{2ax+b}{% \sqrt{4ac-b^{2}}}+C&\,\text{(for }4ac-b^{2}>0\mbox{)}\\ \displaystyle\frac{m}{2a}\ln\left|ax^{2}+bx+c\right|-\frac{2an-bm}{a\sqrt{b^{2% }-4ac}}\,\mathrm{arctanh}\frac{2ax+b}{\sqrt{b^{2}-4ac}}+C&\,\text{(for }4ac-b^% {2}<0\mbox{)}\\ \displaystyle\frac{m}{2a}\ln\left|ax^{2}+bx+c\right|-\frac{2an-bm}{a(2ax+b)}+C% &\,\text{(for }4ac-b^{2}=0\mbox{)}\end{cases}
  25. 1 ( a x 2 + b x + c ) n d x = 2 a x + b ( n - 1 ) ( 4 a c - b 2 ) ( a x 2 + b x + c ) n - 1 + ( 2 n - 3 ) 2 a ( n - 1 ) ( 4 a c - b 2 ) 1 ( a x 2 + b x + c ) n - 1 d x + C \int\frac{1}{(ax^{2}+bx+c)^{n}}\,dx=\frac{2ax+b}{(n-1)(4ac-b^{2})(ax^{2}+bx+c)% ^{n-1}}+\frac{(2n-3)2a}{(n-1)(4ac-b^{2})}\int\frac{1}{(ax^{2}+bx+c)^{n-1}}\,dx+C
  26. x ( a x 2 + b x + c ) n d x = - b x + 2 c ( n - 1 ) ( 4 a c - b 2 ) ( a x 2 + b x + c ) n - 1 - b ( 2 n - 3 ) ( n - 1 ) ( 4 a c - b 2 ) 1 ( a x 2 + b x + c ) n - 1 d x + C \int\frac{x}{(ax^{2}+bx+c)^{n}}\,dx=-\frac{bx+2c}{(n-1)(4ac-b^{2})(ax^{2}+bx+c% )^{n-1}}-\frac{b(2n-3)}{(n-1)(4ac-b^{2})}\int\frac{1}{(ax^{2}+bx+c)^{n-1}}\,dx+C
  27. 1 x ( a x 2 + b x + c ) d x = 1 2 c ln | x 2 a x 2 + b x + c | - b 2 c 1 a x 2 + b x + c d x + C \int\frac{1}{x(ax^{2}+bx+c)}\,dx=\frac{1}{2c}\ln\left|\frac{x^{2}}{ax^{2}+bx+c% }\right|-\frac{b}{2c}\int\frac{1}{ax^{2}+bx+c}\,dx+C
  28. x m ( a + b x n ) p d x = x m + 1 ( a + b x n ) p m + n p + 1 + a n p m + n p + 1 x m ( a + b x n ) p - 1 d x \int x^{m}\left(a+b\,x^{n}\right)^{p}dx=\frac{x^{m+1}\left(a+b\,x^{n}\right)^{% p}}{m+n\,p+1}\,+\,\frac{a\,n\,p}{m+n\,p+1}\int x^{m}\left(a+b\,x^{n}\right)^{p% -1}dx
  29. x m ( a + b x n ) p d x = - x m + 1 ( a + b x n ) p + 1 a n ( p + 1 ) + m + n ( p + 1 ) + 1 a n ( p + 1 ) x m ( a + b x n ) p + 1 d x \int x^{m}\left(a+b\,x^{n}\right)^{p}dx=-\frac{x^{m+1}\left(a+b\,x^{n}\right)^% {p+1}}{a\,n(p+1)}\,+\,\frac{m+n(p+1)+1}{a\,n(p+1)}\int x^{m}\left(a+b\,x^{n}% \right)^{p+1}dx
  30. x m ( a + b x n ) p d x = x m + 1 ( a + b x n ) p m + 1 - b n p m + 1 x m + n ( a + b x n ) p - 1 d x \int x^{m}\left(a+b\,x^{n}\right)^{p}dx=\frac{x^{m+1}\left(a+b\,x^{n}\right)^{% p}}{m+1}\,-\,\frac{b\,n\,p}{m+1}\int x^{m+n}\left(a+b\,x^{n}\right)^{p-1}dx
  31. x m ( a + b x n ) p d x = x m - n + 1 ( a + b x n ) p + 1 b n ( p + 1 ) - m - n + 1 b n ( p + 1 ) x m - n ( a + b x n ) p + 1 d x \int x^{m}\left(a+b\,x^{n}\right)^{p}dx=\frac{x^{m-n+1}\left(a+b\,x^{n}\right)% ^{p+1}}{b\,n(p+1)}\,-\,\frac{m-n+1}{b\,n(p+1)}\int x^{m-n}\left(a+b\,x^{n}% \right)^{p+1}dx
  32. x m ( a + b x n ) p d x = x m - n + 1 ( a + b x n ) p + 1 b ( m + n p + 1 ) - a ( m - n + 1 ) b ( m + n p + 1 ) x m - n ( a + b x n ) p d x \int x^{m}\left(a+b\,x^{n}\right)^{p}dx=\frac{x^{m-n+1}\left(a+b\,x^{n}\right)% ^{p+1}}{b(m+n\,p+1)}\,-\,\frac{a(m-n+1)}{b(m+n\,p+1)}\int x^{m-n}\left(a+b\,x^% {n}\right)^{p}dx
  33. x m ( a + b x n ) p d x = x m + 1 ( a + b x n ) p + 1 a ( m + 1 ) - b ( m + n ( p + 1 ) + 1 ) a ( m + 1 ) x m + n ( a + b x n ) p d x \int x^{m}\left(a+b\,x^{n}\right)^{p}dx=\frac{x^{m+1}\left(a+b\,x^{n}\right)^{% p+1}}{a(m+1)}\,-\,\frac{b(m+n(p+1)+1)}{a(m+1)}\int x^{m+n}\left(a+b\,x^{n}% \right)^{p}dx
  34. ( a + b x ) m ( c + d x ) n ( e + f x ) p (a+b\,x)^{m}(c+d\,x)^{n}(e+f\,x)^{p}
  35. ( A + B x ) ( a + b x ) m ( c + d x ) n ( e + f x ) p d x = - ( A b - a B ) ( a + b x ) m + 1 ( c + d x ) n ( e + f x ) p + 1 b ( m + 1 ) ( a f - b e ) + 1 b ( m + 1 ) ( a f - b e ) \int(A+B\,x)(a+b\,x)^{m}(c+d\,x)^{n}(e+f\,x)^{p}dx=-\frac{(A\,b-a\,B)(a+b\,x)^% {m+1}(c+d\,x)^{n}(e+f\,x)^{p+1}}{b(m+1)(a\,f-b\,e)}\,+\,\frac{1}{b(m+1)(a\,f-b% \,e)}\,\cdot
  36. ( b c ( m + 1 ) ( A f - B e ) + ( A b - a B ) ( n d e + c f ( p + 1 ) ) + d ( b ( m + 1 ) ( A f - B e ) + f ( n + p + 1 ) ( A b - a B ) ) x ) ( a + b x ) m + 1 ( c + d x ) n - 1 ( e + f x ) p d x \int(b\,c(m+1)(A\,f-B\,e)+(A\,b-a\,B)(n\,d\,e+c\,f(p+1))+d(b(m+1)(A\,f-B\,e)+f% (n+p+1)(A\,b-a\,B))x)(a+b\,x)^{m+1}(c+d\,x)^{n-1}(e+f\,x)^{p}dx
  37. ( A + B x ) ( a + b x ) m ( c + d x ) n ( e + f x ) p d x = B ( a + b x ) m ( c + d x ) n + 1 ( e + f x ) p + 1 d f ( m + n + p + 2 ) + 1 d f ( m + n + p + 2 ) \int(A+B\,x)(a+b\,x)^{m}(c+d\,x)^{n}(e+f\,x)^{p}dx=\frac{B(a+b\,x)^{m}(c+d\,x)% ^{n+1}(e+f\,x)^{p+1}}{d\,f(m+n+p+2)}\,+\,\frac{1}{d\,f(m+n+p+2)}\,\cdot
  38. ( A a d f ( m + n + p + 2 ) - B ( b c e m + a ( d e ( n + 1 ) + c f ( p + 1 ) ) ) + ( A b d f ( m + n + p + 2 ) + B ( a d f m - b ( d e ( m + n + 1 ) + c f ( m + p + 1 ) ) ) ) x ) ( a + b x ) m - 1 ( c + d x ) n ( e + f x ) p d x \int(A\,a\,d\,f(m+n+p+2)-B(b\,c\,e\,m+a(d\,e(n+1)+c\,f(p+1)))+(A\,b\,d\,f(m+n+% p+2)+B(a\,d\,f\,m-b(d\,e(m+n+1)+c\,f(m+p+1))))x)(a+b\,x)^{m-1}(c+d\,x)^{n}(e+f% \,x)^{p}dx
  39. ( A + B x ) ( a + b x ) m ( c + d x ) n ( e + f x ) p d x = ( A b - a B ) ( a + b x ) m + 1 ( c + d x ) n + 1 ( e + f x ) p + 1 ( m + 1 ) ( a d - b c ) ( a f - b e ) + 1 ( m + 1 ) ( a d - b c ) ( a f - b e ) \int(A+B\,x)(a+b\,x)^{m}(c+d\,x)^{n}(e+f\,x)^{p}dx=\frac{(A\,b-a\,B)(a+b\,x)^{% m+1}(c+d\,x)^{n+1}(e+f\,x)^{p+1}}{(m+1)(a\,d-b\,c)(a\,f-b\,e)}\,+\,\frac{1}{(m% +1)(a\,d-b\,c)(a\,f-b\,e)}\,\cdot
  40. ( ( m + 1 ) ( A ( a d f - b ( c f + d e ) ) + B b c e ) - ( A b - a B ) ( d e ( n + 1 ) + c f ( p + 1 ) ) - d f ( m + n + p + 3 ) ( A b - a B ) x ) ( a + b x ) m + 1 ( c + d x ) n ( e + f x ) p d x \int((m+1)(A(a\,d\,f-b(c\,f+d\,e))+B\,b\,c\,e)-(A\,b-a\,B)(d\,e(n+1)+c\,f(p+1)% )-d\,f(m+n+p+3)(A\,b-a\,B)x)(a+b\,x)^{m+1}(c+d\,x)^{n}(e+f\,x)^{p}dx
  41. ( a + b x n ) p ( c + d x n ) q \left(a+b\,x^{n}\right)^{p}\left(c+d\,x^{n}\right)^{q}
  42. x m ( a + b x n ) p ( c + d x n ) q x^{m}\left(a+b\,x^{n}\right)^{p}\left(c+d\,x^{n}\right)^{q}
  43. x m ( A + B x n ) ( a + b x n ) p ( c + d x n ) q d x = - ( A b - a B ) x m + 1 ( a + b x n ) p + 1 ( c + d x n ) q a b n ( p + 1 ) + 1 a b n ( p + 1 ) \int x^{m}\left(A+B\,x^{n}\right)\left(a+b\,x^{n}\right)^{p}\left(c+d\,x^{n}% \right)^{q}dx=-\frac{(A\,b-a\,B)x^{m+1}\left(a+b\,x^{n}\right)^{p+1}\left(c+d% \,x^{n}\right)^{q}}{a\,b\,n(p+1)}\,+\,\frac{1}{a\,b\,n(p+1)}\,\cdot
  44. x m ( c ( A b n ( p + 1 ) + ( A b - a B ) ( m + 1 ) ) + d ( A b n ( p + 1 ) + ( A b - a B ) ( m + n q + 1 ) ) x n ) ( a + b x n ) p + 1 ( c + d x n ) q - 1 d x \int x^{m}\left(c(A\,b\,n(p+1)+(A\,b-a\,B)(m+1))+d(A\,b\,n(p+1)+(A\,b-a\,B)(m+% n\,q+1))x^{n}\right)\left(a+b\,x^{n}\right)^{p+1}\left(c+d\,x^{n}\right)^{q-1}dx
  45. x m ( A + B x n ) ( a + b x n ) p ( c + d x n ) q d x = B x m + 1 ( a + b x n ) p + 1 ( c + d x n ) q b ( m + n ( p + q + 1 ) + 1 ) + 1 b ( m + n ( p + q + 1 ) + 1 ) \int x^{m}\left(A+B\,x^{n}\right)\left(a+b\,x^{n}\right)^{p}\left(c+d\,x^{n}% \right)^{q}dx=\frac{B\,x^{m+1}\left(a+b\,x^{n}\right)^{p+1}\left(c+d\,x^{n}% \right)^{q}}{b(m+n(p+q+1)+1)}\,+\,\frac{1}{b(m+n(p+q+1)+1)}\,\cdot
  46. x m ( c ( ( A b - a B ) ( 1 + m ) + A b n ( 1 + p + q ) ) + ( d ( A b - a B ) ( 1 + m ) + B n q ( b c - a d ) + A b d n ( 1 + p + q ) ) x n ) ( a + b x n ) p ( c + d x n ) q - 1 d x \int x^{m}\left(c((A\,b-a\,B)(1+m)+A\,b\,n(1+p+q))+(d(A\,b-a\,B)(1+m)+B\,n\,q(% b\,c-a\,d)+A\,b\,d\,n(1+p+q))\,x^{n}\right)\left(a+b\,x^{n}\right)^{p}\left(c+% d\,x^{n}\right)^{q-1}dx
  47. x m ( A + B x n ) ( a + b x n ) p ( c + d x n ) q d x = - ( A b - a B ) x m + 1 ( a + b x n ) p + 1 ( c + d x n ) q + 1 a n ( b c - a d ) ( p + 1 ) + 1 a n ( b c - a d ) ( p + 1 ) \int x^{m}\left(A+B\,x^{n}\right)\left(a+b\,x^{n}\right)^{p}\left(c+d\,x^{n}% \right)^{q}dx=-\frac{(A\,b-a\,B)x^{m+1}\left(a+b\,x^{n}\right)^{p+1}\left(c+d% \,x^{n}\right)^{q+1}}{a\,n(b\,c-a\,d)(p+1)}\,+\,\frac{1}{a\,n(b\,c-a\,d)(p+1)}\,\cdot
  48. x m ( c ( A b - a B ) ( m + 1 ) + A n ( b c - a d ) ( p + 1 ) + d ( A b - a B ) ( m + n ( p + q + 2 ) + 1 ) x n ) ( a + b x n ) p + 1 ( c + d x n ) q d x \int x^{m}\left(c(A\,b-a\,B)(m+1)+A\,n(b\,c-a\,d)(p+1)+d(A\,b-a\,B)(m+n(p+q+2)% +1)x^{n}\right)\left(a+b\,x^{n}\right)^{p+1}\left(c+d\,x^{n}\right)^{q}dx
  49. x m ( A + B x n ) ( a + b x n ) p ( c + d x n ) q d x = B x m - n + 1 ( a + b x n ) p + 1 ( c + d x n ) q + 1 b d ( m + n ( p + q + 1 ) + 1 ) - 1 b d ( m + n ( p + q + 1 ) + 1 ) \int x^{m}\left(A+B\,x^{n}\right)\left(a+b\,x^{n}\right)^{p}\left(c+d\,x^{n}% \right)^{q}dx=\frac{B\,x^{m-n+1}\left(a+b\,x^{n}\right)^{p+1}\left(c+d\,x^{n}% \right)^{q+1}}{b\,d(m+n(p+q+1)+1)}\,-\,\frac{1}{b\,d(m+n(p+q+1)+1)}\,\cdot
  50. x m - n ( a B c ( m - n + 1 ) + ( a B d ( m + n q + 1 ) - b ( - B c ( m + n p + 1 ) + A d ( m + n ( p + q + 1 ) + 1 ) ) ) x n ) ( a + b x n ) p ( c + d x n ) q d x \int x^{m-n}\left(a\,B\,c(m-n+1)+(a\,B\,d(m+n\,q+1)-b(-B\,c(m+n\,p+1)+A\,d(m+n% (p+q+1)+1)))x^{n}\right)\left(a+b\,x^{n}\right)^{p}\left(c+d\,x^{n}\right)^{q}dx
  51. x m ( A + B x n ) ( a + b x n ) p ( c + d x n ) q d x = A x m + 1 ( a + b x n ) p + 1 ( c + d x n ) q + 1 a c ( m + 1 ) + 1 a c ( m + 1 ) \int x^{m}\left(A+B\,x^{n}\right)\left(a+b\,x^{n}\right)^{p}\left(c+d\,x^{n}% \right)^{q}dx=\frac{A\,x^{m+1}\left(a+b\,x^{n}\right)^{p+1}\left(c+d\,x^{n}% \right)^{q+1}}{a\,c(m+1)}\,+\,\frac{1}{a\,c(m+1)}\,\cdot
  52. x m + n ( a B c ( m + 1 ) - A ( b c + a d ) ( m + n + 1 ) - A n ( b c p + a d q ) - A b d ( m + n ( p + q + 2 ) + 1 ) x n ) ( a + b x n ) p ( c + d x n ) q d x \int x^{m+n}\left(a\,B\,c(m+1)-A(b\,c+a\,d)(m+n+1)-A\,n(b\,c\,p+a\,d\,q)-A\,b% \,d(m+n(p+q+2)+1)x^{n}\right)\left(a+b\,x^{n}\right)^{p}\left(c+d\,x^{n}\right% )^{q}dx
  53. x m ( A + B x n ) ( a + b x n ) p ( c + d x n ) q d x = A x m + 1 ( a + b x n ) p + 1 ( c + d x n ) q a ( m + 1 ) - 1 a ( m + 1 ) \int x^{m}\left(A+B\,x^{n}\right)\left(a+b\,x^{n}\right)^{p}\left(c+d\,x^{n}% \right)^{q}dx=\frac{A\,x^{m+1}\left(a+b\,x^{n}\right)^{p+1}\left(c+d\,x^{n}% \right)^{q}}{a(m+1)}\,-\,\frac{1}{a(m+1)}\,\cdot
  54. x m + n ( c ( A b - a B ) ( m + 1 ) + A n ( b c ( p + 1 ) + a d q ) + d ( ( A b - a B ) ( m + 1 ) + A b n ( p + q + 1 ) ) x n ) ( a + b x n ) p ( c + d x n ) q - 1 d x \int x^{m+n}\left(c(A\,b-a\,B)(m+1)+A\,n(b\,c(p+1)+a\,d\,q)+d((A\,b-a\,B)(m+1)% +A\,b\,n(p+q+1))x^{n}\right)\left(a+b\,x^{n}\right)^{p}\left(c+d\,x^{n}\right)% ^{q-1}dx
  55. x m ( A + B x n ) ( a + b x n ) p ( c + d x n ) q d x = ( A b - a B ) x m - n + 1 ( a + b x n ) p + 1 ( c + d x n ) q + 1 b n ( b c - a d ) ( p + 1 ) - 1 b n ( b c - a d ) ( p + 1 ) \int x^{m}\left(A+B\,x^{n}\right)\left(a+b\,x^{n}\right)^{p}\left(c+d\,x^{n}% \right)^{q}dx=\frac{(A\,b-a\,B)x^{m-n+1}\left(a+b\,x^{n}\right)^{p+1}\left(c+d% \,x^{n}\right)^{q+1}}{b\,n(b\,c-a\,d)(p+1)}\,-\,\frac{1}{b\,n(b\,c-a\,d)(p+1)}\,\cdot
  56. x m - n ( c ( A b - a B ) ( m - n + 1 ) + ( d ( A b - a B ) ( m + n q + 1 ) - b n ( B c - A d ) ( p + 1 ) ) x n ) ( a + b x n ) p + 1 ( c + d x n ) q d x \int x^{m-n}\left(c(A\,b-a\,B)(m-n+1)+(d(A\,b-a\,B)(m+n\,q+1)-b\,n(B\,c-A\,d)(% p+1))x^{n}\right)\left(a+b\,x^{n}\right)^{p+1}\left(c+d\,x^{n}\right)^{q}dx
  57. ( a + b x + c x 2 ) p \left(a+b\,x+c\,x^{2}\right)^{p}
  58. b 2 - 4 a c = 0 b^{2}-4\,a\,c=0
  59. ( d + e x ) m ( a + b x + c x 2 ) p d x = ( d + e x ) m + 1 ( a + b x + c x 2 ) p e ( m + 1 ) - p ( d + e x ) m + 2 ( b + 2 c x ) ( a + b x + c x 2 ) p - 1 e 2 ( m + 1 ) ( m + 2 p + 1 ) + p ( 2 p - 1 ) ( 2 c d - b e ) e 2 ( m + 1 ) ( m + 2 p + 1 ) ( d + e x ) m + 1 ( a + b x + c x 2 ) p - 1 d x \int(d+e\,x)^{m}\left(a+b\,x+c\,x^{2}\right)^{p}dx=\frac{(d+e\,x)^{m+1}\left(a% +b\,x+c\,x^{2}\right)^{p}}{e(m+1)}\,-\,\frac{p(d+e\,x)^{m+2}(b+2c\,x)\left(a+b% \,x+c\,x^{2}\right)^{p-1}}{e^{2}(m+1)(m+2p+1)}\,+\,\frac{p(2p-1)(2c\,d-b\,e)}{% e^{2}(m+1)(m+2p+1)}\int(d+e\,x)^{m+1}\left(a+b\,x+c\,x^{2}\right)^{p-1}dx
  60. ( d + e x ) m ( a + b x + c x 2 ) p d x = ( d + e x ) m + 1 ( a + b x + c x 2 ) p e ( m + 1 ) - p ( d + e x ) m + 2 ( b + 2 c x ) ( a + b x + c x 2 ) p - 1 e 2 ( m + 1 ) ( m + 2 ) + 2 c p ( 2 p - 1 ) e 2 ( m + 1 ) ( m + 2 ) ( d + e x ) m + 2 ( a + b x + c x 2 ) p - 1 d x \int(d+e\,x)^{m}\left(a+b\,x+c\,x^{2}\right)^{p}dx=\frac{(d+e\,x)^{m+1}\left(a% +b\,x+c\,x^{2}\right)^{p}}{e(m+1)}\,-\,\frac{p(d+e\,x)^{m+2}(b+2\,c\,x)\left(a% +b\,x+c\,x^{2}\right)^{p-1}}{e^{2}(m+1)(m+2)}\,+\,\frac{2\,c\,p\,(2\,p-1)}{e^{% 2}(m+1)(m+2)}\int(d+e\,x)^{m+2}\left(a+b\,x+c\,x^{2}\right)^{p-1}dx
  61. ( d + e x ) m ( a + b x + c x 2 ) p d x = - e ( m + 2 p + 2 ) ( d + e x ) m ( a + b x + c x 2 ) p + 1 ( p + 1 ) ( 2 p + 1 ) ( 2 c d - b e ) + ( d + e x ) m + 1 ( b + 2 c x ) ( a + b x + c x 2 ) p ( 2 p + 1 ) ( 2 c d - b e ) + e 2 m ( m + 2 p + 2 ) ( p + 1 ) ( 2 p + 1 ) ( 2 c d - b e ) ( d + e x ) m - 1 ( a + b x + c x 2 ) p + 1 d x \int(d+e\,x)^{m}\left(a+b\,x+c\,x^{2}\right)^{p}dx=-\frac{e(m+2p+2)(d+e\,x)^{m% }\left(a+b\,x+c\,x^{2}\right)^{p+1}}{(p+1)(2p+1)(2c\,d-b\,e)}\,+\,\frac{(d+e\,% x)^{m+1}(b+2c\,x)\left(a+b\,x+c\,x^{2}\right)^{p}}{(2p+1)(2c\,d-b\,e)}\,+\,% \frac{e^{2}m(m+2p+2)}{(p+1)(2p+1)(2c\,d-b\,e)}\int(d+e\,x)^{m-1}\left(a+b\,x+c% \,x^{2}\right)^{p+1}dx
  62. ( d + e x ) m ( a + b x + c x 2 ) p d x = - e m ( d + e x ) m - 1 ( a + b x + c x 2 ) p + 1 2 c ( p + 1 ) ( 2 p + 1 ) + ( d + e x ) m ( b + 2 c x ) ( a + b x + c x 2 ) p 2 c ( 2 p + 1 ) + e 2 m ( m - 1 ) 2 c ( p + 1 ) ( 2 p + 1 ) ( d + e x ) m - 2 ( a + b x + c x 2 ) p + 1 d x \int(d+e\,x)^{m}\left(a+b\,x+c\,x^{2}\right)^{p}dx=-\frac{e\,m(d+e\,x)^{m-1}% \left(a+b\,x+c\,x^{2}\right)^{p+1}}{2c(p+1)(2p+1)}\,+\,\frac{(d+e\,x)^{m}(b+2c% \,x)\left(a+b\,x+c\,x^{2}\right)^{p}}{2c(2p+1)}\,+\,\frac{e^{2}m(m-1)}{2c(p+1)% (2p+1)}\int(d+e\,x)^{m-2}\left(a+b\,x+c\,x^{2}\right)^{p+1}dx
  63. ( d + e x ) m ( a + b x + c x 2 ) p d x = ( d + e x ) m + 1 ( a + b x + c x 2 ) p e ( m + 2 p + 1 ) - p ( 2 c d - b e ) ( d + e x ) m + 1 ( b + 2 c x ) ( a + b x + c x 2 ) p - 1 2 c e 2 ( m + 2 p ) ( m + 2 p + 1 ) + p ( 2 p - 1 ) ( 2 c d - b e ) 2 2 c e 2 ( m + 2 p ) ( m + 2 p + 1 ) ( d + e x ) m ( a + b x + c x 2 ) p - 1 d x \int(d+e\,x)^{m}\left(a+b\,x+c\,x^{2}\right)^{p}dx=\frac{(d+e\,x)^{m+1}\left(a% +b\,x+c\,x^{2}\right)^{p}}{e(m+2p+1)}\,-\,\frac{p(2c\,d-b\,e)(d+e\,x)^{m+1}(b+% 2c\,x)\left(a+b\,x+c\,x^{2}\right)^{p-1}}{2c\,e^{2}(m+2p)(m+2p+1)}\,+\,\frac{p% (2p-1)(2c\,d-b\,e)^{2}}{2c\,e^{2}(m+2p)(m+2p+1)}\int(d+e\,x)^{m}\left(a+b\,x+c% \,x^{2}\right)^{p-1}dx
  64. ( d + e x ) m ( a + b x + c x 2 ) p d x = - 2 c e ( m + 2 p + 2 ) ( d + e x ) m + 1 ( a + b x + c x 2 ) p + 1 ( p + 1 ) ( 2 p + 1 ) ( 2 c d - b e ) 2 + ( d + e x ) m + 1 ( b + 2 c x ) ( a + b x + c x 2 ) p ( 2 p + 1 ) ( 2 c d - b e ) + 2 c e 2 ( m + 2 p + 2 ) ( m + 2 p + 3 ) ( p + 1 ) ( 2 p + 1 ) ( 2 c d - b e ) 2 ( d + e x ) m ( a + b x + c x 2 ) p + 1 d x \int(d+e\,x)^{m}\left(a+b\,x+c\,x^{2}\right)^{p}dx=-\frac{2c\,e(m+2p+2)(d+e\,x% )^{m+1}\left(a+b\,x+c\,x^{2}\right)^{p+1}}{(p+1)(2p+1)(2c\,d-b\,e)^{2}}\,+\,% \frac{(d+e\,x)^{m+1}(b+2c\,x)\left(a+b\,x+c\,x^{2}\right)^{p}}{(2p+1)(2c\,d-b% \,e)}\,+\,\frac{2c\,e^{2}(m+2p+2)(m+2p+3)}{(p+1)(2p+1)(2c\,d-b\,e)^{2}}\int(d+% e\,x)^{m}\left(a+b\,x+c\,x^{2}\right)^{p+1}dx
  65. ( d + e x ) m ( a + b x + c x 2 ) p d x = ( d + e x ) m ( b + 2 c x ) ( a + b x + c x 2 ) p 2 c ( m + 2 p + 1 ) + m ( 2 c d - b e ) 2 c ( m + 2 p + 1 ) ( d + e x ) m - 1 ( a + b x + c x 2 ) p d x \int(d+e\,x)^{m}\left(a+b\,x+c\,x^{2}\right)^{p}dx=\frac{(d+e\,x)^{m}(b+2c\,x)% \left(a+b\,x+c\,x^{2}\right)^{p}}{2c(m+2p+1)}\,+\,\frac{m(2c\,d-b\,e)}{2c(m+2p% +1)}\int(d+e\,x)^{m-1}\left(a+b\,x+c\,x^{2}\right)^{p}dx
  66. ( d + e x ) m ( a + b x + c x 2 ) p d x = - ( d + e x ) m + 1 ( b + 2 c x ) ( a + b x + c x 2 ) p ( m + 1 ) ( 2 c d - b e ) + 2 c ( m + 2 p + 2 ) ( m + 1 ) ( 2 c d - b e ) ( d + e x ) m + 1 ( a + b x + c x 2 ) p d x \int(d+e\,x)^{m}\left(a+b\,x+c\,x^{2}\right)^{p}dx=-\frac{(d+e\,x)^{m+1}(b+2c% \,x)\left(a+b\,x+c\,x^{2}\right)^{p}}{(m+1)(2c\,d-b\,e)}\,+\,\frac{2c(m+2p+2)}% {(m+1)(2c\,d-b\,e)}\int(d+e\,x)^{m+1}\left(a+b\,x+c\,x^{2}\right)^{p}dx
  67. ( a + b x + c x 2 ) p \left(a+b\,x+c\,x^{2}\right)^{p}
  68. ( d + e x ) m ( a + b x + c x 2 ) p (d+e\,x)^{m}\left(a+b\,x+c\,x^{2}\right)^{p}
  69. ( d + e x ) m ( A + B x ) ( a + b x + c x 2 ) p d x = ( d + e x ) m + 1 ( A e ( m + 2 p + 2 ) - B d ( 2 p + 1 ) + e B ( m + 1 ) x ) ( a + b x + c x 2 ) p e 2 ( m + 1 ) ( m + 2 p + 2 ) + 1 e 2 ( m + 1 ) ( m + 2 p + 2 ) p \int(d+e\,x)^{m}(A+B\,x)\left(a+b\,x+c\,x^{2}\right)^{p}dx=\frac{(d+e\,x)^{m+1% }(A\,e(m+2p+2)-B\,d(2p+1)+e\,B(m+1)x)\left(a+b\,x+c\,x^{2}\right)^{p}}{e^{2}(m% +1)(m+2p+2)}\,+\,\frac{1}{e^{2}(m+1)(m+2p+2)}p\,\cdot
  70. ( d + e x ) m + 1 ( B ( b d + 2 a e + 2 a e m + 2 b d p ) - A b e ( m + 2 p + 2 ) + ( B ( 2 c d + b e + b e m + 4 c d p ) - 2 A c e ( m + 2 p + 2 ) ) x ) ( a + b x + c x 2 ) p - 1 d x \int(d+e\,x)^{m+1}(B(b\,d+2a\,e+2a\,e\,m+2b\,d\,p)-A\,b\,e(m+2p+2)+(B(2c\,d+b% \,e+b\,em+4c\,d\,p)-2A\,c\,e(m+2p+2))x)\left(a+b\,x+c\,x^{2}\right)^{p-1}dx
  71. ( d + e x ) m ( A + B x ) ( a + b x + c x 2 ) p d x = ( d + e x ) m ( A b - 2 a B - ( b B - 2 A c ) x ) ( a + b x + c x 2 ) p + 1 ( p + 1 ) ( b 2 - 4 a c ) + 1 ( p + 1 ) ( b 2 - 4 a c ) \int(d+e\,x)^{m}(A+B\,x)\left(a+b\,x+c\,x^{2}\right)^{p}dx=\frac{(d+e\,x)^{m}(% A\,b-2a\,B-(b\,B-2A\,c)x)\left(a+b\,x+c\,x^{2}\right)^{p+1}}{(p+1)\left(b^{2}-% 4a\,c\right)}\,+\,\frac{1}{(p+1)\left(b^{2}-4a\,c\right)}\,\cdot
  72. ( d + e x ) m - 1 ( B ( 2 a e m + b d ( 2 p + 3 ) ) - A ( b e m + 2 c d ( 2 p + 3 ) ) + e ( b B - 2 A c ) ( m + 2 p + 3 ) x ) ( a + b x + c x 2 ) p + 1 d x \int(d+e\,x)^{m-1}(B(2a\,e\,m+b\,d(2p+3))-A(b\,e\,m+2c\,d(2p+3))+e(b\,B-2A\,c)% (m+2p+3)x)\left(a+b\,x+c\,x^{2}\right)^{p+1}dx
  73. ( d + e x ) m ( A + B x ) ( a + b x + c x 2 ) p d x = ( d + e x ) m + 1 ( A c e ( m + 2 p + 2 ) - B ( c d + 2 c d p - b e p ) + B c e ( m + 2 p + 1 ) x ) ( a + b x + c x 2 ) p c e 2 ( m + 2 p + 1 ) ( m + 2 p + 2 ) - p c e 2 ( m + 2 p + 1 ) ( m + 2 p + 2 ) \int(d+e\,x)^{m}(A+B\,x)\left(a+b\,x+c\,x^{2}\right)^{p}dx=\frac{(d+e\,x)^{m+1% }(A\,c\,e(m+2p+2)-B(c\,d+2c\,d\,p-b\,e\,p)+B\,c\,e(m+2p+1)x)\left(a+b\,x+c\,x^% {2}\right)^{p}}{c\,e^{2}(m+2p+1)(m+2p+2)}\,-\,\frac{p}{c\,e^{2}(m+2p+1)(m+2p+2% )}\,\cdot
  74. ( d + e x ) m ( A c e ( b d - 2 a e ) ( m + 2 p + 2 ) + B ( a e ( b e - 2 c d m + b e m ) + b d ( b e p - c d - 2 c d p ) ) + \int(d+e\,x)^{m}(A\,c\,e(b\,d-2a\,e)(m+2p+2)+B(a\,e(b\,e-2c\,d\,m+b\,e\,m)+b\,% d(b\,e\,p-c\,d-2c\,d\,p))+
  75. ( A c e ( 2 c d - b e ) ( m + 2 p + 2 ) - B ( - b 2 e 2 ( m + p + 1 ) + 2 c 2 d 2 ( 1 + 2 p ) + c e ( b d ( m - 2 p ) + 2 a e ( m + 2 p + 1 ) ) ) ) x ) ( a + b x + c x 2 ) p - 1 d x \left(A\,c\,e(2c\,d-b\,e)(m+2p+2)-B\left(-b^{2}e^{2}(m+p+1)+2c^{2}d^{2}(1+2p)+% c\,e(b\,d(m-2p)+2a\,e(m+2p+1))\right)\right)x)\left(a+b\,x+c\,x^{2}\right)^{p-% 1}dx
  76. ( d + e x ) m ( A + B x ) ( a + b x + c x 2 ) p d x = ( d + e x ) m + 1 ( A ( b c d - b 2 e + 2 a c e ) - a B ( 2 c d - b e ) + c ( A ( 2 c d - b e ) - B ( b d - 2 a e ) ) x ) ( a + b x + c x 2 ) p + 1 ( p + 1 ) ( b 2 - 4 a c ) ( c d 2 - b d e + a e 2 ) + \int(d+e\,x)^{m}(A+B\,x)\left(a+b\,x+c\,x^{2}\right)^{p}dx=\frac{(d+e\,x)^{m+1% }\left(A\left(b\,c\,d-b^{2}e+2a\,c\,e\right)-a\,B(2c\,d-b\,e)+c(A(2c\,d-b\,e)-% B(b\,d-2a\,e))x\right)\left(a+b\,x+c\,x^{2}\right)^{p+1}}{(p+1)\left(b^{2}-4a% \,c\right)\left(c\,d^{2}-b\,d\,e+a\,e^{2}\right)}\,+\,
  77. 1 ( p + 1 ) ( b 2 - 4 a c ) ( c d 2 - b d e + a e 2 ) \frac{1}{(p+1)\left(b^{2}-4a\,c\right)\left(c\,d^{2}-b\,d\,e+a\,e^{2}\right)}\,\cdot
  78. ( d + e x ) m ( A ( b c d e ( 2 p - m + 2 ) + b 2 e 2 ( m + p + 2 ) - 2 c 2 d 2 ( 3 + 2 p ) - 2 a c e 2 ( m + 2 p + 3 ) ) - \int(d+e\,x)^{m}(A\left(b\,c\,d\,e(2p-m+2)+b^{2}e^{2}(m+p+2)-2c^{2}d^{2}(3+2p)% -2a\,c\,e^{2}(m+2p+3)\right)-
  79. B ( a e ( b e - 2 c d m + b e m ) + b d ( - 3 c d + b e - 2 c d p + b e p ) ) + c e ( B ( b d - 2 a e ) - A ( 2 c d - b e ) ) ( m + 2 p + 4 ) x ) ( a + b x + c x 2 ) p + 1 d x B(a\,e(b\,e-2c\,dm+b\,e\,m)+b\,d(-3c\,d+b\,e-2c\,d\,p+b\,e\,p))+c\,e(B(b\,d-2a% \,e)-A(2c\,d-b\,e))(m+2p+4)x)\left(a+b\,x+c\,x^{2}\right)^{p+1}dx
  80. ( d + e x ) m ( A + B x ) ( a + b x + c x 2 ) p d x = B ( d + e x ) m ( a + b x + c x 2 ) p + 1 c ( m + 2 p + 2 ) + 1 c ( m + 2 p + 2 ) \int(d+e\,x)^{m}(A+B\,x)\left(a+b\,x+c\,x^{2}\right)^{p}dx=\frac{B(d+e\,x)^{m}% \left(a+b\,x+c\,x^{2}\right)^{p+1}}{c(m+2p+2)}\,+\,\frac{1}{c(m+2p+2)}\,\cdot
  81. ( d + e x ) m - 1 ( m ( A c d - a B e ) - d ( b B - 2 A c ) ( p + 1 ) + ( ( B c d - b B e + A c e ) m - e ( b B - 2 A c ) ( p + 1 ) ) x ) ( a + b x + c x 2 ) p d x \int(d+e\,x)^{m-1}(m(A\,c\,d-a\,B\,e)-d(b\,B-2A\,c)(p+1)+((B\,c\,d-b\,B\,e+A\,% c\,e)m-e(b\,B-2A\,c)(p+1))x)\left(a+b\,x+c\,x^{2}\right)^{p}dx
  82. ( d + e x ) m ( A + B x ) ( a + b x + c x 2 ) p d x = - ( B d - A e ) ( d + e x ) m + 1 ( a + b x + c x 2 ) p + 1 ( m + 1 ) ( c d 2 - b d e + a e 2 ) + 1 ( m + 1 ) ( c d 2 - b d e + a e 2 ) \int(d+e\,x)^{m}(A+B\,x)\left(a+b\,x+c\,x^{2}\right)^{p}dx=-\frac{(B\,d-A\,e)(% d+e\,x)^{m+1}\left(a+b\,x+c\,x^{2}\right)^{p+1}}{(m+1)\left(c\,d^{2}-b\,d\,e+a% \,e^{2}\right)}\,+\,\frac{1}{(m+1)\left(c\,d^{2}-b\,d\,e+a\,e^{2}\right)}\,\cdot
  83. ( d + e x ) m + 1 ( ( A c d - A b e + a B e ) ( m + 1 ) + b ( B d - A e ) ( p + 1 ) + c ( B d - A e ) ( m + 2 p + 3 ) x ) ( a + b x + c x 2 ) p d x \int(d+e\,x)^{m+1}((A\,c\,d-A\,b\,e+a\,B\,e)(m+1)+b(B\,d-A\,e)(p+1)+c(B\,d-A\,% e)(m+2p+3)x)\left(a+b\,x+c\,x^{2}\right)^{p}dx
  84. ( a + b x n + c x 2 n ) p \left(a+b\,x^{n}+c\,x^{2n}\right)^{p}
  85. b 2 - 4 a c = 0 b^{2}-4\,a\,c=0
  86. x m ( a + b x n + c x 2 n ) p d x = x m + 1 ( a + b x n + c x 2 n ) p m + 2 n p + 1 + n p x m + 1 ( 2 a + b x n ) ( a + b x n + c x 2 n ) p - 1 ( m + 1 ) ( m + 2 n p + 1 ) - b n 2 p ( 2 p - 1 ) ( m + 1 ) ( m + 2 n p + 1 ) x m + n ( a + b x n + c x 2 n ) p - 1 d x \int x^{m}\left(a+b\,x^{n}+c\,x^{2n}\right)^{p}dx=\frac{x^{m+1}\left(a+b\,x^{n% }+c\,x^{2n}\right)^{p}}{m+2n\,p+1}\,+\,\frac{n\,p\,x^{m+1}\left(2a+b\,x^{n}% \right)\left(a+b\,x^{n}+c\,x^{2n}\right)^{p-1}}{(m+1)(m+2n\,p+1)}\,-\,\frac{b% \,n^{2}p(2p-1)}{(m+1)(m+2n\,p+1)}\int x^{m+n}\left(a+b\,x^{n}+c\,x^{2n}\right)% ^{p-1}dx
  87. x m ( a + b x n + c x 2 n ) p d x = ( m + n ( 2 p - 1 ) + 1 ) x m + 1 ( a + b x n + c x 2 n ) p ( m + 1 ) ( m + n + 1 ) + n p x m + 1 ( 2 a + b x n ) ( a + b x n + c x 2 n ) p - 1 ( m + 1 ) ( m + n + 1 ) + 2 c p n 2 ( 2 p - 1 ) ( m + 1 ) ( m + n + 1 ) x m + 2 n ( a + b x n + c x 2 n ) p - 1 d x \int x^{m}\left(a+b\,x^{n}+c\,x^{2n}\right)^{p}dx=\frac{(m+n(2p-1)+1)x^{m+1}% \left(a+b\,x^{n}+c\,x^{2n}\right)^{p}}{(m+1)(m+n+1)}\,+\,\frac{n\,p\,x^{m+1}% \left(2a+b\,x^{n}\right)\left(a+b\,x^{n}+c\,x^{2n}\right)^{p-1}}{(m+1)(m+n+1)}% \,+\,\frac{2c\,p\,n^{2}(2p-1)}{(m+1)(m+n+1)}\int x^{m+2n}\left(a+b\,x^{n}+c\,x% ^{2n}\right)^{p-1}dx
  88. x m ( a + b x n + c x 2 n ) p d x = ( m + n ( 2 p + 1 ) + 1 ) x m - n + 1 ( a + b x n + c x 2 n ) p + 1 b n 2 ( p + 1 ) ( 2 p + 1 ) - x m + 1 ( b + 2 c x n ) ( a + b x n + c x 2 n ) p b n ( 2 p + 1 ) - ( m - n + 1 ) ( m + n ( 2 p + 1 ) + 1 ) b n 2 ( p + 1 ) ( 2 p + 1 ) x m - n ( a + b x n + c x 2 n ) p + 1 d x \int x^{m}\left(a+b\,x^{n}+c\,x^{2n}\right)^{p}dx=\frac{(m+n(2p+1)+1)x^{m-n+1}% \left(a+b\,x^{n}+c\,x^{2n}\right)^{p+1}}{b\,n^{2}(p+1)(2p+1)}\,-\,\frac{x^{m+1% }\left(b+2c\,x^{n}\right)\left(a+b\,x^{n}+c\,x^{2n}\right)^{p}}{b\,n(2p+1)}\,-% \,\frac{(m-n+1)(m+n(2p+1)+1)}{b\,n^{2}(p+1)(2p+1)}\int x^{m-n}\left(a+b\,x^{n}% +c\,x^{2n}\right)^{p+1}dx
  89. x m ( a + b x n + c x 2 n ) p d x = - ( m - 3 n - 2 n p + 1 ) x m - 2 n + 1 ( a + b x n + c x 2 n ) p + 1 2 c n 2 ( p + 1 ) ( 2 p + 1 ) - x m - 2 n + 1 ( 2 a + b x n ) ( a + b x n + c x 2 n ) p 2 c n ( 2 p + 1 ) + ( m - n + 1 ) ( m - 2 n + 1 ) 2 c n 2 ( p + 1 ) ( 2 p + 1 ) x m - 2 n ( a + b x n + c x 2 n ) p + 1 d x \int x^{m}\left(a+b\,x^{n}+c\,x^{2n}\right)^{p}dx=-\frac{(m-3n-2n\,p+1)x^{m-2n% +1}\left(a+b\,x^{n}+c\,x^{2n}\right)^{p+1}}{2c\,n^{2}(p+1)(2p+1)}\,-\,\frac{x^% {m-2n+1}\left(2a+b\,x^{n}\right)\left(a+b\,x^{n}+c\,x^{2n}\right)^{p}}{2c\,n(2% p+1)}\,+\,\frac{(m-n+1)(m-2n+1)}{2c\,n^{2}(p+1)(2p+1)}\int x^{m-2n}\left(a+b\,% x^{n}+c\,x^{2n}\right)^{p+1}dx
  90. x m ( a + b x n + c x 2 n ) p d x = x m + 1 ( a + b x n + c x 2 n ) p m + 2 n p + 1 + n p x m + 1 ( 2 a + b x n ) ( a + b x n + c x 2 n ) p - 1 ( m + 2 n p + 1 ) ( m + n ( 2 p - 1 ) + 1 ) + 2 a n 2 p ( 2 p - 1 ) ( m + 2 n p + 1 ) ( m + n ( 2 p - 1 ) + 1 ) x m ( a + b x n + c x 2 n ) p - 1 d x \int x^{m}\left(a+b\,x^{n}+c\,x^{2n}\right)^{p}dx=\frac{x^{m+1}\left(a+b\,x^{n% }+c\,x^{2n}\right)^{p}}{m+2n\,p+1}\,+\,\frac{n\,p\,x^{m+1}\left(2a+b\,x^{n}% \right)\left(a+b\,x^{n}+c\,x^{2n}\right)^{p-1}}{(m+2n\,p+1)(m+n(2p-1)+1)}\,+\,% \frac{2a\,n^{2}p(2p-1)}{(m+2n\,p+1)(m+n(2p-1)+1)}\int x^{m}\left(a+b\,x^{n}+c% \,x^{2n}\right)^{p-1}dx
  91. x m ( a + b x n + c x 2 n ) p d x = - ( m + n + 2 n p + 1 ) x m + 1 ( a + b x n + c x 2 n ) p + 1 2 a n 2 ( p + 1 ) ( 2 p + 1 ) - x m + 1 ( 2 a + b x n ) ( a + b x n + c x 2 n ) p 2 a n ( 2 p + 1 ) + ( m + n ( 2 p + 1 ) + 1 ) ( m + 2 n ( p + 1 ) + 1 ) 2 a n 2 ( p + 1 ) ( 2 p + 1 ) x m ( a + b x n + c x 2 n ) p + 1 d x \int x^{m}\left(a+b\,x^{n}+c\,x^{2n}\right)^{p}dx=-\frac{(m+n+2n\,p+1)x^{m+1}% \left(a+b\,x^{n}+c\,x^{2n}\right)^{p+1}}{2a\,n^{2}(p+1)(2p+1)}\,-\,\frac{x^{m+% 1}\left(2a+b\,x^{n}\right)\left(a+b\,x^{n}+c\,x^{2n}\right)^{p}}{2a\,n(2p+1)}% \,+\,\frac{(m+n(2p+1)+1)(m+2n(p+1)+1)}{2a\,n^{2}(p+1)(2p+1)}\int x^{m}\left(a+% b\,x^{n}+c\,x^{2n}\right)^{p+1}dx
  92. x m ( a + b x n + c x 2 n ) p d x = x m - n + 1 ( b + 2 c x n ) ( a + b x n + c x 2 n ) p 2 c ( m + 2 n p + 1 ) - b ( m - n + 1 ) 2 c ( m + 2 n p + 1 ) x m - n ( a + b x n + c x 2 n ) p d x \int x^{m}\left(a+b\,x^{n}+c\,x^{2n}\right)^{p}dx=\frac{x^{m-n+1}\left(b+2c\,x% ^{n}\right)\left(a+b\,x^{n}+c\,x^{2n}\right)^{p}}{2c(m+2n\,p+1)}\,-\,\frac{b(m% -n+1)}{2c(m+2n\,p+1)}\int x^{m-n}\left(a+b\,x^{n}+c\,x^{2n}\right)^{p}dx
  93. x m ( a + b x n + c x 2 n ) p d x = x m + 1 ( b + 2 c x n ) ( a + b x n + c x 2 n ) p b ( m + 1 ) - 2 c ( m + n ( 2 p + 1 ) + 1 ) b ( m + 1 ) x m + n ( a + b x n + c x 2 n ) p d x \int x^{m}\left(a+b\,x^{n}+c\,x^{2n}\right)^{p}dx=\frac{x^{m+1}\left(b+2c\,x^{% n}\right)\left(a+b\,x^{n}+c\,x^{2n}\right)^{p}}{b(m+1)}\,-\,\frac{2c(m+n(2p+1)% +1)}{b(m+1)}\int x^{m+n}\left(a+b\,x^{n}+c\,x^{2n}\right)^{p}dx
  94. ( a + b x n + c x 2 n ) p \left(a+b\,x^{n}+c\,x^{2n}\right)^{p}
  95. x m ( a + b x n + c x 2 n ) p x^{m}\left(a+b\,x^{n}+c\,x^{2n}\right)^{p}
  96. x m ( A + B x n ) ( a + b x n + c x 2 n ) p d x = x m + 1 ( A ( m + n ( 2 p + 1 ) + 1 ) + B ( m + 1 ) x n ) ( a + b x n + c x 2 n ) p ( m + 1 ) ( m + n ( 2 p + 1 ) + 1 ) + n p ( m + 1 ) ( m + n ( 2 p + 1 ) + 1 ) \int x^{m}\left(A+B\,x^{n}\right)\left(a+b\,x^{n}+c\,x^{2n}\right)^{p}dx=\frac% {x^{m+1}\left(A(m+n(2p+1)+1)+B(m+1)x^{n}\right)\left(a+b\,x^{n}+c\,x^{2n}% \right)^{p}}{(m+1)(m+n(2p+1)+1)}\,+\,\frac{n\,p}{(m+1)(m+n(2p+1)+1)}\,\cdot
  97. x m + n ( 2 a B ( m + 1 ) - A b ( m + n ( 2 p + 1 ) + 1 ) + ( b B ( m + 1 ) - 2 A c ( m + n ( 2 p + 1 ) + 1 ) ) x n ) ( a + b x n + c x 2 n ) p - 1 d x \int x^{m+n}\left(2a\,B(m+1)-A\,b(m+n(2p+1)+1)+(b\,B(m+1)-2\,A\,c(m+n(2p+1)+1)% )x^{n}\right)\left(a+b\,x^{n}+c\,x^{2n}\right)^{p-1}dx
  98. x m ( A + B x n ) ( a + b x n + c x 2 n ) p d x = x m - n + 1 ( A b - 2 a B - ( b B - 2 A c ) x n ) ( a + b x n + c x 2 n ) p + 1 n ( p + 1 ) ( b 2 - 4 a c ) + 1 n ( p + 1 ) ( b 2 - 4 a c ) \int x^{m}\left(A+B\,x^{n}\right)\left(a+b\,x^{n}+c\,x^{2n}\right)^{p}dx=\frac% {x^{m-n+1}\left(A\,b-2a\,B-(b\,B-2A\,c)x^{n}\right)\left(a+b\,x^{n}+c\,x^{2n}% \right)^{p+1}}{n(p+1)\left(b^{2}-4a\,c\right)}\,+\,\frac{1}{n(p+1)\left(b^{2}-% 4a\,c\right)}\,\cdot
  99. x m - n ( ( m - n + 1 ) ( 2 a B - A b ) + ( m + 2 n ( p + 1 ) + 1 ) ( b B - 2 A c ) x n ) ( a + b x n + c x 2 n ) p + 1 d x \int x^{m-n}\left((m-n+1)(2a\,B-A\,b)+(m+2n(p+1)+1)(b\,B-2A\,c)x^{n}\right)% \left(a+b\,x^{n}+c\,x^{2n}\right)^{p+1}dx
  100. x m ( A + B x n ) ( a + b x n + c x 2 n ) p d x = x m + 1 ( b B n p + A c ( m + n ( 2 p + 1 ) + 1 ) + B c ( m + 2 n p + 1 ) x n ) ( a + b x n + c x 2 n ) p c ( m + 2 n p + 1 ) ( m + n ( 2 p + 1 ) + 1 ) + n p c ( m + 2 n p + 1 ) ( m + n ( 2 p + 1 ) + 1 ) \int x^{m}\left(A+B\,x^{n}\right)\left(a+b\,x^{n}+c\,x^{2n}\right)^{p}dx=\frac% {x^{m+1}\left(b\,B\,n\,p+A\,c(m+n(2p+1)+1)+B\,c(m+2n\,p+1)x^{n}\right)\left(a+% b\,x^{n}+c\,x^{2n}\right)^{p}}{c(m+2n\,p+1)(m+n(2p+1)+1)}\,+\,\frac{n\,p}{c(m+% 2n\,p+1)(m+n(2p+1)+1)}\,\cdot
  101. x m ( 2 a A c ( m + n ( 2 p + 1 ) + 1 ) - a b B ( m + 1 ) + ( 2 a B c ( m + 2 n p + 1 ) + A b c ( m + n ( 2 p + 1 ) + 1 ) - b 2 B ( m + n p + 1 ) ) x n ) ( a + b x n + c x 2 n ) p - 1 d x \int x^{m}\left(2a\,A\,c(m+n(2p+1)+1)-a\,b\,B(m+1)+\left(2a\,B\,c(m+2n\,p+1)+A% \,b\,c(m+n(2p+1)+1)-b^{2}B(m+n\,p+1)\right)x^{n}\right)\left(a+b\,x^{n}+c\,x^{% 2n}\right)^{p-1}dx
  102. x m ( A + B x n ) ( a + b x n + c x 2 n ) p d x = - x m + 1 ( A b 2 - a b B - 2 a A c + ( A b - 2 a B ) c x n ) ( a + b x n + c x 2 n ) p + 1 a n ( p + 1 ) ( b 2 - 4 a c ) + 1 a n ( p + 1 ) ( b 2 - 4 a c ) \int x^{m}\left(A+B\,x^{n}\right)\left(a+b\,x^{n}+c\,x^{2n}\right)^{p}dx=-% \frac{x^{m+1}\left(A\,b^{2}-a\,b\,B-2a\,A\,c+(A\,b-2a\,B)c\,x^{n}\right)\left(% a+b\,x^{n}+c\,x^{2n}\right)^{p+1}}{a\,n(p+1)\left(b^{2}-4a\,c\right)}\,+\,% \frac{1}{a\,n(p+1)\left(b^{2}-4a\,c\right)}\,\cdot
  103. x m ( ( m + n ( p + 1 ) + 1 ) A b 2 - a b B ( m + 1 ) - 2 ( m + 2 n ( p + 1 ) + 1 ) a A c + ( m + n ( 2 p + 3 ) + 1 ) ( A b - 2 a B ) c x n ) ( a + b x n + c x 2 n ) p + 1 d x \int x^{m}\left((m+n(p+1)+1)A\,b^{2}-a\,b\,B(m+1)-2(m+2n(p+1)+1)a\,A\,c+(m+n(2% p+3)+1)(A\,b-2a\,B)c\,x^{n}\right)\left(a+b\,x^{n}+c\,x^{2n}\right)^{p+1}dx
  104. x m ( A + B x n ) ( a + b x n + c x 2 n ) p d x = B x m - n + 1 ( a + b x n + c x 2 n ) p + 1 c ( m + n ( 2 p + 1 ) + 1 ) - 1 c ( m + n ( 2 p + 1 ) + 1 ) \int x^{m}\left(A+B\,x^{n}\right)\left(a+b\,x^{n}+c\,x^{2n}\right)^{p}dx=\frac% {B\,x^{m-n+1}\left(a+b\,x^{n}+c\,x^{2n}\right)^{p+1}}{c(m+n(2p+1)+1)}\,-\,% \frac{1}{c(m+n(2p+1)+1)}\,\cdot
  105. x m - n ( a B ( m - n + 1 ) + ( b B ( m + n p + 1 ) - A c ( m + n ( 2 p + 1 ) + 1 ) ) x n ) ( a + b x n + c x 2 n ) p d x \int x^{m-n}\left(a\,B(m-n+1)+(b\,B(m+n\,p+1)-A\,c(m+n(2p+1)+1))x^{n}\right)% \left(a+b\,x^{n}+c\,x^{2n}\right)^{p}dx
  106. x m ( A + B x n ) ( a + b x n + c x 2 n ) p d x = A x m + 1 ( a + b x n + c x 2 n ) p + 1 a ( m + 1 ) + 1 a ( m + 1 ) \int x^{m}\left(A+B\,x^{n}\right)\left(a+b\,x^{n}+c\,x^{2n}\right)^{p}dx=\frac% {A\,x^{m+1}\left(a+b\,x^{n}+c\,x^{2n}\right)^{p+1}}{a(m+1)}\,+\,\frac{1}{a(m+1% )}\,\cdot
  107. x m + n ( a B ( m + 1 ) - A b ( m + n ( p + 1 ) + 1 ) - A c ( m + 2 n ( p + 1 ) + 1 ) x n ) ( a + b x n + c x 2 n ) p d x \int x^{m+n}\left(a\,B(m+1)-A\,b(m+n(p+1)+1)-A\,c(m+2n(p+1)+1)x^{n}\right)% \left(a+b\,x^{n}+c\,x^{2n}\right)^{p}dx

List_of_integrals_of_trigonometric_functions.html

  1. sin ( x ) \sin(x)
  2. cos ( x ) \cos(x)
  3. a cos n x d x = a n sin n x + C \int a\cos nx\;\mathrm{d}x=\frac{a}{n}\sin nx+C
  4. sin a x d x = - 1 a cos a x + C \int\sin ax\;\mathrm{d}x=-\frac{1}{a}\cos ax+C\,\!
  5. sin 2 a x d x = x 2 - 1 4 a sin 2 a x + C = x 2 - 1 2 a sin a x cos a x + C \int\sin^{2}{ax}\;\mathrm{d}x=\frac{x}{2}-\frac{1}{4a}\sin 2ax+C=\frac{x}{2}-% \frac{1}{2a}\sin ax\cos ax+C\!
  6. sin 3 a x d x = cos 3 a x 12 a - 3 cos a x 4 a + C \int\sin^{3}{ax}\;\mathrm{d}x=\frac{\cos 3ax}{12a}-\frac{3\cos ax}{4a}+C\!
  7. x sin 2 a x d x = x 2 4 - x 4 a sin 2 a x - 1 8 a 2 cos 2 a x + C \int x\sin^{2}{ax}\;\mathrm{d}x=\frac{x^{2}}{4}-\frac{x}{4a}\sin 2ax-\frac{1}{% 8a^{2}}\cos 2ax+C\!
  8. x 2 sin 2 a x d x = x 3 6 - ( x 2 4 a - 1 8 a 3 ) sin 2 a x - x 4 a 2 cos 2 a x + C \int x^{2}\sin^{2}{ax}\;\mathrm{d}x=\frac{x^{3}}{6}-\left(\frac{x^{2}}{4a}-% \frac{1}{8a^{3}}\right)\sin 2ax-\frac{x}{4a^{2}}\cos 2ax+C\!
  9. sin b 1 x sin b 2 x d x = sin ( ( b 2 - b 1 ) x ) 2 ( b 2 - b 1 ) - sin ( ( b 1 + b 2 ) x ) 2 ( b 1 + b 2 ) + C (for | b 1 | | b 2 | ) \int\sin b_{1}x\sin b_{2}x\;\mathrm{d}x=\frac{\sin((b_{2}-b_{1})x)}{2(b_{2}-b_% {1})}-\frac{\sin((b_{1}+b_{2})x)}{2(b_{1}+b_{2})}+C\qquad\mbox{(for }~{}|b_{1}% |\neq|b_{2}|\mbox{)}~{}\,\!
  10. sin n a x d x = - sin n - 1 a x cos a x n a + n - 1 n sin n - 2 a x d x (for n > 0 ) \int\sin^{n}{ax}\;\mathrm{d}x=-\frac{\sin^{n-1}ax\cos ax}{na}+\frac{n-1}{n}% \int\sin^{n-2}ax\;\mathrm{d}x\qquad\mbox{(for }~{}n>0\mbox{)}~{}\,\!
  11. d x sin a x = - 1 a ln | csc a x + cot a x | + C \int\frac{\mathrm{d}x}{\sin ax}=-\frac{1}{a}\ln{\left|\csc{ax}+\cot{ax}\right|% }+C
  12. d x sin n a x = cos a x a ( 1 - n ) sin n - 1 a x + n - 2 n - 1 d x sin n - 2 a x (for n > 1 ) \int\frac{\mathrm{d}x}{\sin^{n}ax}=\frac{\cos ax}{a(1-n)\sin^{n-1}ax}+\frac{n-% 2}{n-1}\int\frac{\mathrm{d}x}{\sin^{n-2}ax}\qquad\mbox{(for }~{}n>1\mbox{)}~{}\,\!
  13. x sin a x d x = sin a x a 2 - x cos a x a + C \int x\sin ax\;\mathrm{d}x=\frac{\sin ax}{a^{2}}-\frac{x\cos ax}{a}+C\,\!
  14. x n sin a x d x = - x n a cos a x + n a x n - 1 cos a x d x = k = 0 2 k n ( - 1 ) k + 1 x n - 2 k a 1 + 2 k n ! ( n - 2 k ) ! cos a x + k = 0 2 k + 1 n ( - 1 ) k x n - 1 - 2 k a 2 + 2 k n ! ( n - 2 k - 1 ) ! sin a x (for n > 0 ) \int x^{n}\sin ax\;\mathrm{d}x=-\frac{x^{n}}{a}\cos ax+\frac{n}{a}\int x^{n-1}% \cos ax\;\mathrm{d}x=\sum_{k=0}^{2k\leq n}(-1)^{k+1}\frac{x^{n-2k}}{a^{1+2k}}% \frac{n!}{(n-2k)!}\cos ax+\sum_{k=0}^{2k+1\leq n}(-1)^{k}\frac{x^{n-1-2k}}{a^{% 2+2k}}\frac{n!}{(n-2k-1)!}\sin ax\qquad\mbox{(for }~{}n>0\mbox{)}~{}\,\!
  15. sin a x x d x = n = 0 ( - 1 ) n ( a x ) 2 n + 1 ( 2 n + 1 ) ( 2 n + 1 ) ! + C \int\frac{\sin ax}{x}\mathrm{d}x=\sum_{n=0}^{\infty}(-1)^{n}\frac{(ax)^{2n+1}}% {(2n+1)\cdot(2n+1)!}+C\,\!
  16. sin a x x n d x = - sin a x ( n - 1 ) x n - 1 + a n - 1 cos a x x n - 1 d x \int\frac{\sin ax}{x^{n}}\mathrm{d}x=-\frac{\sin ax}{(n-1)x^{n-1}}+\frac{a}{n-% 1}\int\frac{\cos ax}{x^{n-1}}\mathrm{d}x\,\!
  17. d x 1 ± sin a x = 1 a tan ( a x 2 π 4 ) + C \int\frac{\mathrm{d}x}{1\pm\sin ax}=\frac{1}{a}\tan\left(\frac{ax}{2}\mp\frac{% \pi}{4}\right)+C
  18. x d x 1 + sin a x = x a tan ( a x 2 - π 4 ) + 2 a 2 ln | cos ( a x 2 - π 4 ) | + C \int\frac{x\;\mathrm{d}x}{1+\sin ax}=\frac{x}{a}\tan\left(\frac{ax}{2}-\frac{% \pi}{4}\right)+\frac{2}{a^{2}}\ln\left|\cos\left(\frac{ax}{2}-\frac{\pi}{4}% \right)\right|+C
  19. x d x 1 - sin a x = x a cot ( π 4 - a x 2 ) + 2 a 2 ln | sin ( π 4 - a x 2 ) | + C \int\frac{x\;\mathrm{d}x}{1-\sin ax}=\frac{x}{a}\cot\left(\frac{\pi}{4}-\frac{% ax}{2}\right)+\frac{2}{a^{2}}\ln\left|\sin\left(\frac{\pi}{4}-\frac{ax}{2}% \right)\right|+C
  20. sin a x d x 1 ± sin a x = ± x + 1 a tan ( π 4 a x 2 ) + C \int\frac{\sin ax\;\mathrm{d}x}{1\pm\sin ax}=\pm x+\frac{1}{a}\tan\left(\frac{% \pi}{4}\mp\frac{ax}{2}\right)+C
  21. cos a x d x = 1 a sin a x + C \int\cos ax\;\mathrm{d}x=\frac{1}{a}\sin ax+C\,\!
  22. cos 2 a x d x = x 2 + 1 4 a sin 2 a x + C = x 2 + 1 2 a sin a x cos a x + C \int\cos^{2}{ax}\;\mathrm{d}x=\frac{x}{2}+\frac{1}{4a}\sin 2ax+C=\frac{x}{2}+% \frac{1}{2a}\sin ax\cos ax+C\!
  23. cos n a x d x = cos n - 1 a x sin a x n a + n - 1 n cos n - 2 a x d x (for n > 0 ) \int\cos^{n}ax\;\mathrm{d}x=\frac{\cos^{n-1}ax\sin ax}{na}+\frac{n-1}{n}\int% \cos^{n-2}ax\;\mathrm{d}x\qquad\mbox{(for }~{}n>0\mbox{)}~{}\,\!
  24. x cos a x d x = cos a x a 2 + x sin a x a + C \int x\cos ax\;\mathrm{d}x=\frac{\cos ax}{a^{2}}+\frac{x\sin ax}{a}+C\,\!
  25. x 2 cos 2 a x d x = x 3 6 + ( x 2 4 a - 1 8 a 3 ) sin 2 a x + x 4 a 2 cos 2 a x + C \int x^{2}\cos^{2}{ax}\;\mathrm{d}x=\frac{x^{3}}{6}+\left(\frac{x^{2}}{4a}-% \frac{1}{8a^{3}}\right)\sin 2ax+\frac{x}{4a^{2}}\cos 2ax+C\!
  26. x n cos a x d x = x n sin a x a - n a x n - 1 sin a x d x = k = 0 2 k + 1 n ( - 1 ) k x n - 2 k - 1 a 2 + 2 k n ! ( n - 2 k - 1 ) ! cos a x + k = 0 2 k n ( - 1 ) k x n - 2 k a 1 + 2 k n ! ( n - 2 k ) ! sin a x \int x^{n}\cos ax\;\mathrm{d}x=\frac{x^{n}\sin ax}{a}-\frac{n}{a}\int x^{n-1}% \sin ax\;\mathrm{d}x\,=\sum_{k=0}^{2k+1\leq n}(-1)^{k}\frac{x^{n-2k-1}}{a^{2+2% k}}\frac{n!}{(n-2k-1)!}\cos ax+\sum_{k=0}^{2k\leq n}(-1)^{k}\frac{x^{n-2k}}{a^% {1+2k}}\frac{n!}{(n-2k)!}\sin ax\!
  27. cos a x x d x = ln | a x | + k = 1 ( - 1 ) k ( a x ) 2 k 2 k ( 2 k ) ! + C \int\frac{\cos ax}{x}\mathrm{d}x=\ln|ax|+\sum_{k=1}^{\infty}(-1)^{k}\frac{(ax)% ^{2k}}{2k\cdot(2k)!}+C\,\!
  28. cos a x x n d x = - cos a x ( n - 1 ) x n - 1 - a n - 1 sin a x x n - 1 d x (for n 1 ) \int\frac{\cos ax}{x^{n}}\mathrm{d}x=-\frac{\cos ax}{(n-1)x^{n-1}}-\frac{a}{n-% 1}\int\frac{\sin ax}{x^{n-1}}\mathrm{d}x\qquad\mbox{(for }~{}n\neq 1\mbox{)}~{% }\,\!
  29. d x cos a x = 1 a ln | tan ( a x 2 + π 4 ) | + C \int\frac{\mathrm{d}x}{\cos ax}=\frac{1}{a}\ln\left|\tan\left(\frac{ax}{2}+% \frac{\pi}{4}\right)\right|+C
  30. d x cos n a x = sin a x a ( n - 1 ) cos n - 1 a x + n - 2 n - 1 d x cos n - 2 a x (for n > 1 ) \int\frac{\mathrm{d}x}{\cos^{n}ax}=\frac{\sin ax}{a(n-1)\cos^{n-1}ax}+\frac{n-% 2}{n-1}\int\frac{\mathrm{d}x}{\cos^{n-2}ax}\qquad\mbox{(for }~{}n>1\mbox{)}~{}\,\!
  31. d x 1 + cos a x = 1 a tan a x 2 + C \int\frac{\mathrm{d}x}{1+\cos ax}=\frac{1}{a}\tan\frac{ax}{2}+C\,\!
  32. d x 1 - cos a x = - 1 a cot a x 2 + C \int\frac{\mathrm{d}x}{1-\cos ax}=-\frac{1}{a}\cot\frac{ax}{2}+C
  33. x d x 1 + cos a x = x a tan a x 2 + 2 a 2 ln | cos a x 2 | + C \int\frac{x\;\mathrm{d}x}{1+\cos ax}=\frac{x}{a}\tan\frac{ax}{2}+\frac{2}{a^{2% }}\ln\left|\cos\frac{ax}{2}\right|+C
  34. x d x 1 - cos a x = - x a cot a x 2 + 2 a 2 ln | sin a x 2 | + C \int\frac{x\;\mathrm{d}x}{1-\cos ax}=-\frac{x}{a}\cot\frac{ax}{2}+\frac{2}{a^{% 2}}\ln\left|\sin\frac{ax}{2}\right|+C
  35. cos a x d x 1 + cos a x = x - 1 a tan a x 2 + C \int\frac{\cos ax\;\mathrm{d}x}{1+\cos ax}=x-\frac{1}{a}\tan\frac{ax}{2}+C\,\!
  36. cos a x d x 1 - cos a x = - x - 1 a cot a x 2 + C \int\frac{\cos ax\;\mathrm{d}x}{1-\cos ax}=-x-\frac{1}{a}\cot\frac{ax}{2}+C\,\!
  37. cos a 1 x cos a 2 x d x = sin ( ( a 2 - a 1 ) x ) 2 ( a 2 - a 1 ) + sin ( ( a 2 + a 1 ) x ) 2 ( a 2 + a 1 ) + C (for | a 1 | | a 2 | ) \int\cos a_{1}x\cos a_{2}x\;\mathrm{d}x=\frac{\sin((a_{2}-a_{1})x)}{2(a_{2}-a_% {1})}+\frac{\sin((a_{2}+a_{1})x)}{2(a_{2}+a_{1})}+C\qquad\mbox{(for }~{}|a_{1}% |\neq|a_{2}|\mbox{)}~{}\,\!
  38. tan a x d x = - 1 a ln | cos a x | + C = 1 a ln | sec a x | + C \int\tan ax\;\mathrm{d}x=-\frac{1}{a}\ln|\cos ax|+C=\frac{1}{a}\ln|\sec ax|+C\,\!
  39. tan 2 x d x = tan x - x + C \int\tan^{2}{x}\,\mathrm{d}x=\tan{x}-x+C
  40. tan n a x d x = 1 a ( n - 1 ) tan n - 1 a x - tan n - 2 a x d x (for n 1 ) \int\tan^{n}ax\;\mathrm{d}x=\frac{1}{a(n-1)}\tan^{n-1}ax-\int\tan^{n-2}ax\;% \mathrm{d}x\qquad\mbox{(for }~{}n\neq 1\mbox{)}~{}\,\!
  41. d x q tan a x + p = 1 p 2 + q 2 ( p x + q a ln | q sin a x + p cos a x | ) + C (for p 2 + q 2 0 ) \int\frac{\mathrm{d}x}{q\tan ax+p}=\frac{1}{p^{2}+q^{2}}(px+\frac{q}{a}\ln|q% \sin ax+p\cos ax|)+C\qquad\mbox{(for }~{}p^{2}+q^{2}\neq 0\mbox{)}~{}\,\!
  42. d x tan a x + 1 = x 2 + 1 2 a ln | sin a x + cos a x | + C \int\frac{\mathrm{d}x}{\tan ax+1}=\frac{x}{2}+\frac{1}{2a}\ln|\sin ax+\cos ax|% +C\,\!
  43. d x tan a x - 1 = - x 2 + 1 2 a ln | sin a x - cos a x | + C \int\frac{\mathrm{d}x}{\tan ax-1}=-\frac{x}{2}+\frac{1}{2a}\ln|\sin ax-\cos ax% |+C\,\!
  44. tan a x d x tan a x + 1 = x 2 - 1 2 a ln | sin a x + cos a x | + C \int\frac{\tan ax\;\mathrm{d}x}{\tan ax+1}=\frac{x}{2}-\frac{1}{2a}\ln|\sin ax% +\cos ax|+C\,\!
  45. tan a x d x tan a x - 1 = x 2 + 1 2 a ln | sin a x - cos a x | + C \int\frac{\tan ax\;\mathrm{d}x}{\tan ax-1}=\frac{x}{2}+\frac{1}{2a}\ln|\sin ax% -\cos ax|+C\,\!
  46. sec a x d x = 1 a ln | sec a x + tan a x | + C \int\sec{ax}\,\mathrm{d}x=\frac{1}{a}\ln{\left|\sec{ax}+\tan{ax}\right|}+C
  47. sec 2 x d x = tan x + C \int\sec^{2}{x}\,\mathrm{d}x=\tan{x}+C
  48. sec 3 x d x = 1 2 sec x tan x + 1 2 ln | sec x + tan x | + C . \int\sec^{3}x\,dx=\frac{1}{2}\sec x\tan x+\frac{1}{2}\ln|\sec x+\tan x|+C.
  49. sec n a x d x = sec n - 2 a x tan a x a ( n - 1 ) + n - 2 n - 1 sec n - 2 a x d x (for n 1 ) \int\sec^{n}{ax}\,\mathrm{d}x=\frac{\sec^{n-2}{ax}\tan{ax}}{a(n-1)}\,+\,\frac{% n-2}{n-1}\int\sec^{n-2}{ax}\,\mathrm{d}x\qquad\mbox{ (for }~{}n\neq 1\mbox{)}~% {}\,\!
  50. d x sec x + 1 = x - tan x 2 + C \int\frac{\mathrm{d}x}{\sec{x}+1}=x-\tan{\frac{x}{2}}+C
  51. csc ( a x ) d x = - 1 a ln | csc a x + cot a x | + C \int\csc(ax)\mathrm{d}x=-\frac{1}{a}\ln{\left|\csc{ax}+\cot{ax}\right|}+C
  52. csc 2 x d x = - cot x + C \int\csc^{2}{x}\,\mathrm{d}x=-\cot{x}+C
  53. csc n a x d x = - csc n - 1 ( a x ) cos ( a x ) a ( n - 1 ) + n - 2 n - 1 csc n - 2 a x d x (for n 1 ) \int\csc^{n}{ax}\,\mathrm{d}x=-\frac{\csc^{n-1}\left(ax\right)\cos\left(ax% \right)}{a(n-1)}\,+\,\frac{n-2}{n-1}\int\csc^{n-2}{ax}\,\mathrm{d}x\qquad\mbox% { (for }~{}n\neq 1\mbox{)}~{}\,\!
  54. d x csc x + 1 = x - 2 sin x 2 cos x 2 + sin x 2 + C \int\frac{\mathrm{d}x}{\csc{x}+1}=x-\frac{2\sin{\frac{x}{2}}}{\cos{\frac{x}{2}% }+\sin{\frac{x}{2}}}+C
  55. d x csc x - 1 = 2 sin x 2 cos x 2 - sin x 2 - x + C \int\frac{\mathrm{d}x}{\csc{x}-1}=\frac{2\sin{\frac{x}{2}}}{\cos{\frac{x}{2}}-% \sin{\frac{x}{2}}}-x+C
  56. cot a x d x = 1 a ln | sin a x | + C \int\cot ax\;\mathrm{d}x=\frac{1}{a}\ln|\sin ax|+C\,\!
  57. cot n a x d x = - 1 a ( n - 1 ) cot n - 1 a x - cot n - 2 a x d x (for n 1 ) \int\cot^{n}ax\;\mathrm{d}x=-\frac{1}{a(n-1)}\cot^{n-1}ax-\int\cot^{n-2}ax\;% \mathrm{d}x\qquad\mbox{(for }~{}n\neq 1\mbox{)}~{}\,\!
  58. d x 1 + cot a x = tan a x d x tan a x + 1 \int\frac{\mathrm{d}x}{1+\cot ax}=\int\frac{\tan ax\;\mathrm{d}x}{\tan ax+1}\,\!
  59. d x 1 - cot a x = tan a x d x tan a x - 1 \int\frac{\mathrm{d}x}{1-\cot ax}=\int\frac{\tan ax\;\mathrm{d}x}{\tan ax-1}\,\!
  60. d x cos a x ± sin a x = 1 a 2 ln | tan ( a x 2 ± π 8 ) | + C \int\frac{\mathrm{d}x}{\cos ax\pm\sin ax}=\frac{1}{a\sqrt{2}}\ln\left|\tan% \left(\frac{ax}{2}\pm\frac{\pi}{8}\right)\right|+C
  61. d x ( cos a x ± sin a x ) 2 = 1 2 a tan ( a x π 4 ) + C \int\frac{\mathrm{d}x}{(\cos ax\pm\sin ax)^{2}}=\frac{1}{2a}\tan\left(ax\mp% \frac{\pi}{4}\right)+C
  62. d x ( cos x + sin x ) n = 1 n - 1 ( sin x - cos x ( cos x + sin x ) n - 1 - 2 ( n - 2 ) d x ( cos x + sin x ) n - 2 ) \int\frac{\mathrm{d}x}{(\cos x+\sin x)^{n}}=\frac{1}{n-1}\left(\frac{\sin x-% \cos x}{(\cos x+\sin x)^{n-1}}-2(n-2)\int\frac{\mathrm{d}x}{(\cos x+\sin x)^{n% -2}}\right)
  63. cos a x d x cos a x + sin a x = x 2 + 1 2 a ln | sin a x + cos a x | + C \int\frac{\cos ax\;\mathrm{d}x}{\cos ax+\sin ax}=\frac{x}{2}+\frac{1}{2a}\ln% \left|\sin ax+\cos ax\right|+C
  64. cos a x d x cos a x - sin a x = x 2 - 1 2 a ln | sin a x - cos a x | + C \int\frac{\cos ax\;\mathrm{d}x}{\cos ax-\sin ax}=\frac{x}{2}-\frac{1}{2a}\ln% \left|\sin ax-\cos ax\right|+C
  65. sin a x d x cos a x + sin a x = x 2 - 1 2 a ln | sin a x + cos a x | + C \int\frac{\sin ax\;\mathrm{d}x}{\cos ax+\sin ax}=\frac{x}{2}-\frac{1}{2a}\ln% \left|\sin ax+\cos ax\right|+C
  66. sin a x d x cos a x - sin a x = - x 2 - 1 2 a ln | sin a x - cos a x | + C \int\frac{\sin ax\;\mathrm{d}x}{\cos ax-\sin ax}=-\frac{x}{2}-\frac{1}{2a}\ln% \left|\sin ax-\cos ax\right|+C
  67. cos a x d x sin a x ( 1 + cos a x ) = - 1 4 a tan 2 a x 2 + 1 2 a ln | tan a x 2 | + C \int\frac{\cos ax\;\mathrm{d}x}{\sin ax(1+\cos ax)}=-\frac{1}{4a}\tan^{2}\frac% {ax}{2}+\frac{1}{2a}\ln\left|\tan\frac{ax}{2}\right|+C
  68. cos a x d x sin a x ( 1 - cos a x ) = - 1 4 a cot 2 a x 2 - 1 2 a ln | tan a x 2 | + C \int\frac{\cos ax\;\mathrm{d}x}{\sin ax(1-\cos ax)}=-\frac{1}{4a}\cot^{2}\frac% {ax}{2}-\frac{1}{2a}\ln\left|\tan\frac{ax}{2}\right|+C
  69. sin a x d x cos a x ( 1 + sin a x ) = 1 4 a cot 2 ( a x 2 + π 4 ) + 1 2 a ln | tan ( a x 2 + π 4 ) | + C \int\frac{\sin ax\;\mathrm{d}x}{\cos ax(1+\sin ax)}=\frac{1}{4a}\cot^{2}\left(% \frac{ax}{2}+\frac{\pi}{4}\right)+\frac{1}{2a}\ln\left|\tan\left(\frac{ax}{2}+% \frac{\pi}{4}\right)\right|+C
  70. sin a x d x cos a x ( 1 - sin a x ) = 1 4 a tan 2 ( a x 2 + π 4 ) - 1 2 a ln | tan ( a x 2 + π 4 ) | + C \int\frac{\sin ax\;\mathrm{d}x}{\cos ax(1-\sin ax)}=\frac{1}{4a}\tan^{2}\left(% \frac{ax}{2}+\frac{\pi}{4}\right)-\frac{1}{2a}\ln\left|\tan\left(\frac{ax}{2}+% \frac{\pi}{4}\right)\right|+C
  71. sin a x cos a x d x = - 1 2 a cos 2 a x + C \int\sin ax\cos ax\;\mathrm{d}x=-\frac{1}{2a}\cos^{2}ax+C\,\!
  72. sin a 1 x cos a 2 x d x = - cos ( ( a 1 - a 2 ) x ) 2 ( a 1 - a 2 ) - cos ( ( a 1 + a 2 ) x ) 2 ( a 1 + a 2 ) + C (for | a 1 | | a 2 | ) \int\sin a_{1}x\cos a_{2}x\;\mathrm{d}x=-\frac{\cos((a_{1}-a_{2})x)}{2(a_{1}-a% _{2})}-\frac{\cos((a_{1}+a_{2})x)}{2(a_{1}+a_{2})}+C\qquad\mbox{(for }~{}|a_{1% }|\neq|a_{2}|\mbox{)}~{}\,\!
  73. sin n a x cos a x d x = 1 a ( n + 1 ) sin n + 1 a x + C (for n - 1 ) \int\sin^{n}ax\cos ax\;\mathrm{d}x=\frac{1}{a(n+1)}\sin^{n+1}ax+C\qquad\mbox{(% for }~{}n\neq-1\mbox{)}~{}\,\!
  74. sin a x cos n a x d x = - 1 a ( n + 1 ) cos n + 1 a x + C (for n - 1 ) \int\sin ax\cos^{n}ax\;\mathrm{d}x=-\frac{1}{a(n+1)}\cos^{n+1}ax+C\qquad\mbox{% (for }~{}n\neq-1\mbox{)}~{}\,\!
  75. sin n a x cos m a x d x = - sin n - 1 a x cos m + 1 a x a ( n + m ) + n - 1 n + m sin n - 2 a x cos m a x d x (for m , n > 0 ) \int\sin^{n}ax\cos^{m}ax\;\mathrm{d}x=-\frac{\sin^{n-1}ax\cos^{m+1}ax}{a(n+m)}% +\frac{n-1}{n+m}\int\sin^{n-2}ax\cos^{m}ax\;\mathrm{d}x\qquad\mbox{(for }~{}m,% n>0\mbox{)}~{}\,\!
  76. sin n a x cos m a x d x = sin n + 1 a x cos m - 1 a x a ( n + m ) + m - 1 n + m sin n a x cos m - 2 a x d x (for m , n > 0 ) \int\sin^{n}ax\cos^{m}ax\;\mathrm{d}x=\frac{\sin^{n+1}ax\cos^{m-1}ax}{a(n+m)}+% \frac{m-1}{n+m}\int\sin^{n}ax\cos^{m-2}ax\;\mathrm{d}x\qquad\mbox{(for }~{}m,n% >0\mbox{)}~{}\,\!
  77. d x sin a x cos a x = 1 a ln | tan a x | + C \int\frac{\mathrm{d}x}{\sin ax\cos ax}=\frac{1}{a}\ln\left|\tan ax\right|+C
  78. d x sin a x cos n a x = 1 a ( n - 1 ) cos n - 1 a x + d x sin a x cos n - 2 a x (for n 1 ) \int\frac{\mathrm{d}x}{\sin ax\cos^{n}ax}=\frac{1}{a(n-1)\cos^{n-1}ax}+\int% \frac{\mathrm{d}x}{\sin ax\cos^{n-2}ax}\qquad\mbox{(for }~{}n\neq 1\mbox{)}~{}\,\!
  79. d x sin n a x cos a x = - 1 a ( n - 1 ) sin n - 1 a x + d x sin n - 2 a x cos a x (for n 1 ) \int\frac{\mathrm{d}x}{\sin^{n}ax\cos ax}=-\frac{1}{a(n-1)\sin^{n-1}ax}+\int% \frac{\mathrm{d}x}{\sin^{n-2}ax\cos ax}\qquad\mbox{(for }~{}n\neq 1\mbox{)}~{}\,\!
  80. sin a x d x cos n a x = 1 a ( n - 1 ) cos n - 1 a x + C (for n 1 ) \int\frac{\sin ax\;\mathrm{d}x}{\cos^{n}ax}=\frac{1}{a(n-1)\cos^{n-1}ax}+C% \qquad\mbox{(for }~{}n\neq 1\mbox{)}~{}\,\!
  81. sin 2 a x d x cos a x = - 1 a sin a x + 1 a ln | tan ( π 4 + a x 2 ) | + C \int\frac{\sin^{2}ax\;\mathrm{d}x}{\cos ax}=-\frac{1}{a}\sin ax+\frac{1}{a}\ln% \left|\tan\left(\frac{\pi}{4}+\frac{ax}{2}\right)\right|+C
  82. sin 2 a x d x cos n a x = sin a x a ( n - 1 ) cos n - 1 a x - 1 n - 1 d x cos n - 2 a x (for n 1 ) \int\frac{\sin^{2}ax\;\mathrm{d}x}{\cos^{n}ax}=\frac{\sin ax}{a(n-1)\cos^{n-1}% ax}-\frac{1}{n-1}\int\frac{\mathrm{d}x}{\cos^{n-2}ax}\qquad\mbox{(for }~{}n% \neq 1\mbox{)}~{}\,\!
  83. sin n a x d x cos a x = - sin n - 1 a x a ( n - 1 ) + sin n - 2 a x d x cos a x (for n 1 ) \int\frac{\sin^{n}ax\;\mathrm{d}x}{\cos ax}=-\frac{\sin^{n-1}ax}{a(n-1)}+\int% \frac{\sin^{n-2}ax\;\mathrm{d}x}{\cos ax}\qquad\mbox{(for }~{}n\neq 1\mbox{)}~% {}\,\!
  84. sin n a x d x cos m a x = sin n + 1 a x a ( m - 1 ) cos m - 1 a x - n - m + 2 m - 1 sin n a x d x cos m - 2 a x (for m 1 ) \int\frac{\sin^{n}ax\;\mathrm{d}x}{\cos^{m}ax}=\frac{\sin^{n+1}ax}{a(m-1)\cos^% {m-1}ax}-\frac{n-m+2}{m-1}\int\frac{\sin^{n}ax\;\mathrm{d}x}{\cos^{m-2}ax}% \qquad\mbox{(for }~{}m\neq 1\mbox{)}~{}\,\!
  85. sin n a x d x cos m a x = - sin n - 1 a x a ( n - m ) cos m - 1 a x + n - 1 n - m sin n - 2 a x d x cos m a x (for m n ) \int\frac{\sin^{n}ax\;\mathrm{d}x}{\cos^{m}ax}=-\frac{\sin^{n-1}ax}{a(n-m)\cos% ^{m-1}ax}+\frac{n-1}{n-m}\int\frac{\sin^{n-2}ax\;\mathrm{d}x}{\cos^{m}ax}% \qquad\mbox{(for }~{}m\neq n\mbox{)}~{}\,\!
  86. sin n a x d x cos m a x = sin n - 1 a x a ( m - 1 ) cos m - 1 a x - n - 1 m - 1 sin n - 2 a x d x cos m - 2 a x (for m 1 ) \int\frac{\sin^{n}ax\;\mathrm{d}x}{\cos^{m}ax}=\frac{\sin^{n-1}ax}{a(m-1)\cos^% {m-1}ax}-\frac{n-1}{m-1}\int\frac{\sin^{n-2}ax\;\mathrm{d}x}{\cos^{m-2}ax}% \qquad\mbox{(for }~{}m\neq 1\mbox{)}~{}\,\!
  87. cos a x d x sin n a x = - 1 a ( n - 1 ) sin n - 1 a x + C (for n 1 ) \int\frac{\cos ax\;\mathrm{d}x}{\sin^{n}ax}=-\frac{1}{a(n-1)\sin^{n-1}ax}+C% \qquad\mbox{(for }~{}n\neq 1\mbox{)}~{}\,\!
  88. cos 2 a x d x sin a x = 1 a ( cos a x + ln | tan a x 2 | ) + C \int\frac{\cos^{2}ax\;\mathrm{d}x}{\sin ax}=\frac{1}{a}\left(\cos ax+\ln\left|% \tan\frac{ax}{2}\right|\right)+C
  89. cos 2 a x d x sin n a x = - 1 n - 1 ( cos a x a sin n - 1 a x + d x sin n - 2 a x ) (for n 1 ) \int\frac{\cos^{2}ax\;\mathrm{d}x}{\sin^{n}ax}=-\frac{1}{n-1}\left(\frac{\cos ax% }{a\sin^{n-1}ax}+\int\frac{\mathrm{d}x}{\sin^{n-2}ax}\right)\qquad\mbox{(for }% ~{}n\neq 1\mbox{)}~{}
  90. cos n a x d x sin m a x = - cos n + 1 a x a ( m - 1 ) sin m - 1 a x - n - m + 2 m - 1 cos n a x d x sin m - 2 a x (for m 1 ) \int\frac{\cos^{n}ax\;\mathrm{d}x}{\sin^{m}ax}=-\frac{\cos^{n+1}ax}{a(m-1)\sin% ^{m-1}ax}-\frac{n-m+2}{m-1}\int\frac{\cos^{n}ax\;\mathrm{d}x}{\sin^{m-2}ax}% \qquad\mbox{(for }~{}m\neq 1\mbox{)}~{}\,\!
  91. cos n a x d x sin m a x = cos n - 1 a x a ( n - m ) sin m - 1 a x + n - 1 n - m cos n - 2 a x d x sin m a x (for m n ) \int\frac{\cos^{n}ax\;\mathrm{d}x}{\sin^{m}ax}=\frac{\cos^{n-1}ax}{a(n-m)\sin^% {m-1}ax}+\frac{n-1}{n-m}\int\frac{\cos^{n-2}ax\;\mathrm{d}x}{\sin^{m}ax}\qquad% \mbox{(for }~{}m\neq n\mbox{)}~{}\,\!
  92. cos n a x d x sin m a x = - cos n - 1 a x a ( m - 1 ) sin m - 1 a x - n - 1 m - 1 cos n - 2 a x d x sin m - 2 a x (for m 1 ) \int\frac{\cos^{n}ax\;\mathrm{d}x}{\sin^{m}ax}=-\frac{\cos^{n-1}ax}{a(m-1)\sin% ^{m-1}ax}-\frac{n-1}{m-1}\int\frac{\cos^{n-2}ax\;\mathrm{d}x}{\sin^{m-2}ax}% \qquad\mbox{(for }~{}m\neq 1\mbox{)}~{}\,\!
  93. sin a x tan a x d x = 1 a ( ln | sec a x + tan a x | - sin a x ) + C \int\sin ax\tan ax\;\mathrm{d}x=\frac{1}{a}(\ln|\sec ax+\tan ax|-\sin ax)+C\,\!
  94. tan n a x d x sin 2 a x = 1 a ( n - 1 ) tan n - 1 ( a x ) + C (for n 1 ) \int\frac{\tan^{n}ax\;\mathrm{d}x}{\sin^{2}ax}=\frac{1}{a(n-1)}\tan^{n-1}(ax)+% C\qquad\mbox{(for }~{}n\neq 1\mbox{)}~{}\,\!
  95. tan n a x d x cos 2 a x = 1 a ( n + 1 ) tan n + 1 a x + C (for n - 1 ) \int\frac{\tan^{n}ax\;\mathrm{d}x}{\cos^{2}ax}=\frac{1}{a(n+1)}\tan^{n+1}ax+C% \qquad\mbox{(for }~{}n\neq-1\mbox{)}~{}\,\!
  96. cot n a x d x sin 2 a x = - 1 a ( n + 1 ) cot n + 1 a x + C (for n - 1 ) \int\frac{\cot^{n}ax\;\mathrm{d}x}{\sin^{2}ax}=-\frac{1}{a(n+1)}\cot^{n+1}ax+C% \qquad\mbox{(for }~{}n\neq-1\mbox{)}~{}\,\!
  97. cot n a x d x cos 2 a x = 1 a ( 1 - n ) tan 1 - n a x + C (for n 1 ) \int\frac{\cot^{n}ax\;\mathrm{d}x}{\cos^{2}ax}=\frac{1}{a(1-n)}\tan^{1-n}ax+C% \qquad\mbox{(for }~{}n\neq 1\mbox{)}~{}\,\!
  98. sec x tan x d x = sec x + C \int\sec x\tan x\;\mathrm{d}x=\sec x+C
  99. < m t p l > 0 π 2 sin n x d x = 0 π 2 cos n x d x = { n - 1 n n - 3 n - 2 3 4 1 2 π 2 , if n is even n - 1 n n - 3 n - 2 4 5 2 3 if n is odd and more than 1 \int_{<}mtpl>{{0}}^{{\frac{\pi}{2}}}\sin^{n}x\,dx=\int_{{0}}^{{\frac{\pi}{2}}}% \cos^{n}x\,dx=\begin{cases}\frac{n-1}{n}\cdot\frac{n-3}{n-2}\cdots\frac{3}{4}% \cdot\frac{1}{2}\cdot\frac{\pi}{2},&\,\text{if }n\,\text{ is even}\\ \frac{n-1}{n}\cdot\frac{n-3}{n-2}\cdots\frac{4}{5}\cdot\frac{2}{3}&\,\text{if % }n\,\text{ is odd and more than 1}\end{cases}
  100. < m t p l > - c c sin x d x = 0 \int_{<}mtpl>{{-c}}^{{c}}\sin{x}\;\mathrm{d}x=0\!
  101. < m t p l > - c c cos x d x = 2 0 c cos x d x = 2 - c 0 cos x d x = 2 sin c \int_{<}mtpl>{{-c}}^{{c}}\cos{x}\;\mathrm{d}x=2\int_{{0}}^{{c}}\cos{x}\;% \mathrm{d}x=2\int_{{-c}}^{{0}}\cos{x}\;\mathrm{d}x=2\sin{c}\!
  102. < m t p l > - c c tan x d x = 0 \int_{<}mtpl>{{-c}}^{{c}}\tan{x}\;\mathrm{d}x=0\!
  103. - a 2 a 2 x 2 cos 2 n π x a d x = a 3 ( n 2 π 2 - 6 ) 24 n 2 π 2 (for n = 1 , 3 , 5... ) \int_{-\frac{a}{2}}^{\frac{a}{2}}x^{2}\cos^{2}{\frac{n\pi x}{a}}\;\mathrm{d}x=% \frac{a^{3}(n^{2}\pi^{2}-6)}{24n^{2}\pi^{2}}\qquad\mbox{(for }~{}n=1,3,5...% \mbox{)}~{}\,\!
  104. - a 2 a 2 x 2 sin 2 n π x a d x = a 3 ( n 2 π 2 - 6 ( - 1 ) n ) 24 n 2 π 2 = a 3 24 ( 1 - 6 ( - 1 ) n n 2 π 2 ) (for n = 1 , 2 , 3 , ) \int_{\frac{-a}{2}}^{\frac{a}{2}}x^{2}\sin^{2}{\frac{n\pi x}{a}}\;\mathrm{d}x=% \frac{a^{3}(n^{2}\pi^{2}-6(-1)^{n})}{24n^{2}\pi^{2}}=\frac{a^{3}}{24}(1-6\frac% {(-1)^{n}}{n^{2}\pi^{2}})\qquad\mbox{(for }~{}n=1,2,3,...\mbox{)}~{}\,\!
  105. < m t p l > 0 2 π sin 2 m + 1 x cos 2 n + 1 x d x = 0 { n , m } \int_{<}mtpl>{{0}}^{{2\pi}}\sin^{2m+1}{x}\cos^{2n+1}{x}\;\mathrm{d}x=0\!\qquad% \{n,m\}\in\mathbb{Z}

List_of_mathematical_symbols.html

  1. \doteq
  2. cosh x := e x + e - x 2 \cosh x:=\frac{e^{x}+e^{-x}}{2}

List_of_matrices.html

  1. I n = [ 1 0 0 0 1 0 0 0 1 ] . I_{n}=\begin{bmatrix}1&0&\cdots&0\\ 0&1&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&1\end{bmatrix}.
  2. f ( x ) = x T A x f(x)=x^{T}Ax
  3. ( C ) i , j = r = 1 n A i , r B r , j . (C)_{i,j}=\sum_{r=1}^{n}A_{i,r}B_{r,j}.
  4. A - ( y T A x ) - 1 A x y T A A-(y^{T}Ax)^{-1}Axy^{T}A

List_of_numbers.html

  1. 10 4 2 20 {10}^{\,\!4\cdot 2^{20}}
  2. 10 4 2 30 {10}^{\,\!4\cdot 2^{30}}
  3. 10 4 2 40 {10}^{\,\!4\cdot 2^{40}}
  4. 10 4 2 50 {10}^{\,\!4\cdot 2^{50}}
  5. 10 4 2 60 {10}^{\,\!4\cdot 2^{60}}
  6. 10 4 2 70 {10}^{\,\!4\cdot 2^{70}}
  7. 10 4 2 80 {10}^{\,\!4\cdot 2^{80}}
  8. 10 4 2 90 {10}^{\,\!4\cdot 2^{90}}
  9. 10 4 2 100 {10}^{\,\!4\cdot 2^{100}}
  10. 10 4 2 1000 {10}^{\,\!4\cdot 2^{1000}}
  11. 10 4 2 10 , 000 {10}^{\,\!4\cdot 2^{10,000}}
  12. ( 3 25 ) \left({3\over 25}\right)
  13. ( 9 75 ) \left({9\over 75}\right)
  14. ( 6 50 ) \left({6\over 50}\right)
  15. ( 12 100 ) \left({12\over 100}\right)
  16. ( 24 200 ) \left({24\over 200}\right)
  17. 1 1 1\over 1
  18. 9 10 9\over 10
  19. 4 5 4\over 5
  20. 7 10 7\over 10
  21. 3 5 3\over 5
  22. 1 2 1\over 2
  23. 2 5 2\over 5
  24. 1 3 1\over 3
  25. 3 10 3\over 10
  26. 1 4 1\over 4
  27. 1 5 1\over 5
  28. 1 6 1\over 6
  29. 1 7 1\over 7
  30. 1 8 1\over 8
  31. 1 9 1\over 9
  32. 1 10 1\over 10
  33. 1 11 1\over 11
  34. 9 100 9\over 100
  35. 1 12 1\over 12
  36. 2 25 2\over 25
  37. 1 16 1\over 16
  38. 1 20 1\over 20
  39. 1 21 1\over 21
  40. 1 22 1\over 22
  41. 1 23 1\over 23
  42. 1 30 1\over 30
  43. 1 60 1\over 60
  44. 1 81 1\over 81
  45. 1 100 1\over 100
  46. 1 1000 1\over 1000
  47. 1 3600 1\over 3600
  48. 1 10000 1\over 10000
  49. 1 10 5 1\over 10^{5}
  50. 1 10 6 1\over 10^{6}
  51. 1 10 7 1\over 10^{7}
  52. 1 10 8 1\over 10^{8}
  53. 1 10 9 1\over 10^{9}
  54. 0 1 0\over 1
  55. 3 4 \frac{\sqrt{3}}{4}
  56. 5 - 1 2 {\sqrt{5}-1}\over 2
  57. Φ \Phi\,
  58. 3 2 \frac{\sqrt{3}}{2}
  59. 2 12 \sqrt[12]{2}
  60. 3 2 4 \frac{3\sqrt{2}}{4}
  61. 2 3 \sqrt[3]{2}
  62. 1 2 + 1 6 23 3 3 + \sqrt[3]{\frac{1}{2}+\frac{1}{6}\sqrt{\frac{23}{3}}}+
  63. 1 2 - 1 6 23 3 3 \sqrt[3]{\frac{1}{2}-\frac{1}{6}\sqrt{\frac{23}{3}}}
  64. x 3 = x + 1 . x^{3}=x+1\,.
  65. 2 \sqrt{2}
  66. 2 = 2 sin 45 = 2 cos 45 \sqrt{2}=2\sin 45^{\circ}=2\cos 45^{\circ}
  67. 1 3 + 2 3 116 + 12 93 3 + 1 6 116 + 12 93 3 \frac{1}{3}+\frac{2}{3\sqrt[3]{116+12\sqrt{93}}}+\frac{1}{6}\sqrt[3]{116+12% \sqrt{93}}
  68. 5 + 2 5 2 \frac{\sqrt{5+2\sqrt{5}}}{2}
  69. 17 - 1 2 \frac{\sqrt{17}-1}{2}
  70. 5 + 1 2 {\sqrt{5}+1}\over 2
  71. ( ϕ ) \left(\phi\right)
  72. x 2 = x + 1 . x^{2}=x+1\,.
  73. 5 4 5 - 2 5 \frac{5}{4\sqrt{5-2\sqrt{5}}}
  74. 3 \sqrt{3}
  75. 3 = 2 sin 60 = 2 cos 30 \sqrt{3}=2\sin 60^{\circ}=2\cos 30^{\circ}
  76. 1 × 2 1\times\sqrt{2}
  77. 1 + 19 + 3 33 3 + 19 - 3 33 3 3 \frac{1+\sqrt[3]{19+3\sqrt{33}}+\sqrt[3]{19-3\sqrt{33}}}{3}
  78. 5 \sqrt{5}
  79. 1 × 2 1\times 2
  80. 2 × 3 \sqrt{2}\times\sqrt{3}
  81. 1 × 2 × 2 1\times\sqrt{2}\times\sqrt{2}
  82. 2 + 1 \sqrt{2}+1
  83. ( δ S ) \left(\delta_{S}\right)
  84. x 2 = 2 x + 1 x^{2}=2x+1\,
  85. 6 \sqrt{6}
  86. 2 3 \sqrt{2}\cdot\sqrt{3}
  87. 2 × 3 \sqrt{2}\times\sqrt{3}
  88. 1 × 1 × 2 1\times 1\times 2
  89. 1 × 5 1\times\sqrt{5}
  90. 2 × 2 2\times\sqrt{2}
  91. 3 \sqrt{3}
  92. 3 3 2 \frac{3\sqrt{3}}{2}
  93. 7 \sqrt{7}
  94. 1 × 2 × 2 1\times 2\times\sqrt{2}
  95. 1 × 6 1\times\sqrt{6}
  96. 2 × 3 2\times\sqrt{3}
  97. 2 × 5 \sqrt{2}\times\sqrt{5}
  98. 8 \sqrt{8}
  99. 2 2 2\sqrt{2}
  100. 2 \sqrt{2}
  101. 1 × 7 1\times\sqrt{7}
  102. 2 × 6 \sqrt{2}\times\sqrt{6}
  103. 3 × 5 \sqrt{3}\times\sqrt{5}
  104. 10 \sqrt{10}
  105. 2 5 \sqrt{2}\cdot\sqrt{5}
  106. 2 × 5 \sqrt{2}\times\sqrt{5}
  107. 1 × 3 1\times 3
  108. 2 × 6 2\times\sqrt{6}
  109. 3 × 7 \sqrt{3}\times\sqrt{7}
  110. 5 \sqrt{5}
  111. 11 \sqrt{11}
  112. 1 × 1 × 3 1\times 1\times 3
  113. 1 × 10 1\times\sqrt{10}
  114. 2 × 7 2\times\sqrt{7}
  115. 3 × 2 3\times\sqrt{2}
  116. 3 × 8 \sqrt{3}\times\sqrt{8}
  117. 5 × 6 \sqrt{5}\times\sqrt{6}
  118. 12 \sqrt{12}
  119. 2 3 2\sqrt{3}
  120. 1 × 11 1\times\sqrt{11}
  121. 2 × 8 2\times\sqrt{8}
  122. 3 × 3 3\times\sqrt{3}
  123. 2 × 10 \sqrt{2}\times\sqrt{10}
  124. 5 × 7 \sqrt{5}\times\sqrt{7}
  125. 6 \sqrt{6}
  126. 1 / π {1}/{π}
  127. 1 / e {1}/{e}
  128. i = - 1 i=\sqrt{-1}
  129. ξ n k = cos ( 2 π k n ) + i sin ( 2 π k n ) \xi^{k}_{n}=\cos\left(2\pi\tfrac{k}{n}\right)+i\sin\left(2\pi\tfrac{k}{n}\right)
  130. \infty
  131. 0 \aleph_{0}
  132. \mathbb{N}
  133. 1 \aleph_{1}
  134. 1 \beth_{1}
  135. ( 2 0 ) (2^{\aleph_{0}})
  136. 𝔠 \mathfrak{c}
  137. ( 2 0 ) (2^{\aleph_{0}})
  138. A {}_{A}
  139. × 10 2 3 \times 10^{2}3
  140. k e {k}_{e}
  141. × 10 9 \times 10^{9}
  142. × 10 19 \times 10^{–}19
  143. r {}_{r}
  144. u {}_{u}
  145. × 10 34 \times 10^{–}34
  146. {}_{∞}
  147. × 10 - 8 \times 10^{-}8

List_of_relativistic_equations.html

  1. γ = 1 1 - β 2 \gamma=\frac{1}{\sqrt{1-\beta^{2}}}
  2. t = γ t t^{\prime}=\gamma t
  3. t = 2 h c t=\frac{2h}{c}
  4. c 2 ( t 2 ) 2 = h 2 + v 2 ( t 2 ) 2 c^{2}\left(\frac{t^{\prime}}{2}\right)^{2}=h^{2}+v^{2}\left(\frac{t^{\prime}}{% 2}\right)^{2}
  5. t t^{\prime}
  6. ( t 2 ) 2 = h 2 c 2 - v 2 \left(\frac{t^{\prime}}{2}\right)^{2}=\frac{h^{2}}{c^{2}-v^{2}}
  7. t 2 = h c 2 - v 2 \frac{t^{\prime}}{2}=\frac{h}{\sqrt{c^{2}-v^{2}}}
  8. t = 2 h c 2 - v 2 t^{\prime}=\frac{2h}{\sqrt{c^{2}-v^{2}}}
  9. t = 2 h c 1 1 - v 2 c 2 = t 1 - v 2 c 2 t^{\prime}=\frac{2h}{c}\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}=\frac{t}{\sqrt{1% -\frac{v^{2}}{c^{2}}}}
  10. t = γ t t^{\prime}=\gamma t
  11. = γ \ell^{\prime}=\frac{\ell}{\gamma}
  12. = v t \ell^{\prime}=vt^{\prime}\,
  13. = v t = v ( t γ ) = γ \ell=vt=v\left(\frac{t^{\prime}}{\gamma}\right)=\frac{\ell^{\prime}}{\gamma}
  14. x = γ ( x - v t ) x^{\prime}=\gamma\left(x-vt\right)
  15. y = y y^{\prime}=y\,
  16. z = z z^{\prime}=z\,
  17. t = γ ( t - v x c 2 ) t^{\prime}=\gamma\left(t-\frac{vx}{c^{2}}\right)
  18. x γ = x - v t \frac{x^{\prime}}{\gamma}=x-vt
  19. x = γ ( x - v t ) x^{\prime}=\gamma\left(x-vt\right)
  20. x = γ ( x + v t ) x=\gamma\left(x^{\prime}+vt^{\prime}\right)
  21. y = y y^{\prime}=y\,
  22. z = z z^{\prime}=z\,
  23. x = γ ( γ ( x - v t ) + v t ) x=\gamma\left(\gamma\left(x-vt\right)+vt^{\prime}\right)
  24. x = γ ( γ x - γ v t + v t ) x=\gamma\left(\gamma x-\gamma vt+vt^{\prime}\right)
  25. x = γ 2 x - γ 2 v t + γ v t x=\gamma^{2}x-\gamma^{2}vt+\gamma vt^{\prime}\,
  26. γ v t = γ 2 v t - γ 2 x + x \gamma vt^{\prime}=\gamma^{2}vt-\gamma^{2}x+x\,
  27. γ v t = γ 2 v t + x ( 1 - γ 2 ) \gamma vt^{\prime}=\gamma^{2}vt+x\left(1-\gamma^{2}\right)
  28. γ v t = γ 2 v t + x ( 1 - 1 1 - β 2 ) \gamma vt^{\prime}=\gamma^{2}vt+x\left(1-\frac{1}{1-\beta^{2}}\right)
  29. γ v t = γ 2 v t + x ( 1 - β 2 1 - β 2 - 1 1 - β 2 ) \gamma vt^{\prime}=\gamma^{2}vt+x\left(\frac{1-\beta^{2}}{1-\beta^{2}}-\frac{1% }{1-\beta^{2}}\right)
  30. γ v t = γ 2 v t - x ( β 2 1 - β 2 ) \gamma vt^{\prime}=\gamma^{2}vt-x\left(\frac{\beta^{2}}{1-\beta^{2}}\right)
  31. γ v t = γ 2 v t - γ 2 β 2 x \gamma vt^{\prime}=\gamma^{2}vt-\gamma^{2}\beta^{2}x\,
  32. t = γ ( t - β x c ) t^{\prime}=\gamma\left(t-\beta\frac{x}{c}\right)
  33. t = γ ( t - v x c 2 ) t^{\prime}=\gamma\left(t-\frac{vx}{c^{2}}\right)
  34. x = γ ( x - v t ) x^{\prime}=\gamma\left(x-vt\right)
  35. y = y y^{\prime}=y\,
  36. z = z z^{\prime}=z\,
  37. t = γ ( t - v x c 2 ) t^{\prime}=\gamma\left(t-\frac{vx}{c^{2}}\right)
  38. V x = V x - v 1 - V x v c 2 V^{\prime}_{x}=\frac{V_{x}-v}{1-\frac{V_{x}v}{c^{2}}}
  39. V y = V y γ ( 1 - V x v c 2 ) V^{\prime}_{y}=\frac{V_{y}}{\gamma\left(1-\frac{V_{x}v}{c^{2}}\right)}
  40. V z = V z γ ( 1 - V x v c 2 ) V^{\prime}_{z}=\frac{V_{z}}{\gamma\left(1-\frac{V_{x}v}{c^{2}}\right)}
  41. d x = γ ( d x - v d t ) dx^{\prime}=\gamma\left(dx-vdt\right)
  42. d y = d y dy^{\prime}=dy\,
  43. d z = d z dz^{\prime}=dz\,
  44. d t = γ ( d t - v d x c 2 ) dt^{\prime}=\gamma\left(dt-\frac{vdx}{c^{2}}\right)
  45. d x d t = γ ( d x - v d t ) γ ( d t - v d x c 2 ) \frac{dx^{\prime}}{dt^{\prime}}=\frac{\gamma\left(dx-vdt\right)}{\gamma\left(% dt-\frac{vdx}{c^{2}}\right)}
  46. d x d t = d x - v d t d t - v d x c 2 \frac{dx^{\prime}}{dt^{\prime}}=\frac{dx-vdt}{dt-\frac{vdx}{c^{2}}}
  47. d x d t = d x d t - v 1 - d x d t v c 2 \frac{dx^{\prime}}{dt^{\prime}}=\frac{\frac{dx}{dt}-v}{1-\frac{dx}{dt}\frac{v}% {c^{2}}}
  48. V x = d x d t V x = d x d t V_{x}=\frac{dx}{dt}\,\quad V^{\prime}_{x}=\frac{dx^{\prime}}{dt^{\prime}}
  49. V x = V x - v 1 - V x v c 2 V^{\prime}_{x}=\frac{V_{x}-v}{1-\frac{V_{x}v}{c^{2}}}
  50. V y = V y γ ( 1 - V x v c 2 ) V^{\prime}_{y}=\frac{V_{y}}{\gamma\left(1-\frac{V_{x}v}{c^{2}}\right)}
  51. V z = V z γ ( 1 - V x v c 2 ) V^{\prime}_{z}=\frac{V_{z}}{\gamma\left(1-\frac{V_{x}v}{c^{2}}\right)}
  52. s y m b o l 𝖺 s y m b o l 𝖻 = η ( s y m b o l 𝖺 , s y m b o l 𝖻 ) symbol{\mathsf{a}}\cdot symbol{\mathsf{b}}=\eta(symbol{\mathsf{a}},symbol{% \mathsf{b}})
  53. η \eta
  54. η = ( - 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) \eta=\begin{pmatrix}-1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{pmatrix}
  55. d s 2 = d x 2 + d y 2 + d z 2 - c 2 d t 2 = ( c d t d x d y d z ) ( - 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) ( c d t d x d y d z ) ds^{2}=dx^{2}+dy^{2}+dz^{2}-c^{2}dt^{2}=\begin{pmatrix}cdt&dx&dy&dz\end{% pmatrix}\begin{pmatrix}-1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{pmatrix}\begin{pmatrix}cdt\\ dx\\ dy\\ dz\end{pmatrix}
  56. η ( s y m b o l 𝖺 , s y m b o l 𝖻 ) = η ( Λ s y m b o l 𝖺 , Λ s y m b o l 𝖻 ) = η ( s y m b o l 𝖺 , s y m b o l 𝖻 ) \eta(symbol{\mathsf{a}}^{\prime},symbol{\mathsf{b}}^{\prime})=\eta\left(% \Lambda symbol{\mathsf{a}},\Lambda symbol{\mathsf{b}}\right)=\eta(symbol{% \mathsf{a}},symbol{\mathsf{b}})
  57. x = γ x - γ β c t x^{\prime}=\gamma x-\gamma\beta ct\,
  58. y = y y^{\prime}=y\,
  59. z = z z^{\prime}=z\,
  60. c t = γ c t - γ β x ct^{\prime}=\gamma ct-\gamma\beta x\,
  61. ( c t x y z ) = ( γ - γ β 0 0 - γ β γ 0 0 0 0 1 0 0 0 0 1 ) ( c t x y z ) \begin{pmatrix}ct^{\prime}\\ x^{\prime}\\ y^{\prime}\\ z^{\prime}\end{pmatrix}=\begin{pmatrix}\gamma&-\gamma\beta&0&0\\ -\gamma\beta&\gamma&0&0\\ 0&0&1&0\\ 0&0&0&1\end{pmatrix}\begin{pmatrix}ct\\ x\\ y\\ z\end{pmatrix}
  62. s y m b o l 𝖺 = Λ s y m b o l 𝖺 symbol{\mathsf{a}}^{\prime}=\Lambda symbol{\mathsf{a}}
  63. s y m b o l 𝖺 symbol{\mathsf{a}}^{\prime}
  64. s y m b o l 𝖺 symbol{\mathsf{a}}
  65. s y m b o l 𝖺 symbol{\mathsf{a}}^{\prime}
  66. 𝐫 𝐫 r 2 x 1 2 + x 2 2 + x 3 2 \mathbf{r}\cdot\mathbf{r}\equiv r^{2}\equiv x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\,\!
  67. s y m b o l 𝖷 s y m b o l 𝖷 = ( c τ ) 2 symbol{\mathsf{X}}\cdot symbol{\mathsf{X}}=\left(c\tau\right)^{2}\,\!
  68. ( c t ) 2 - ( x 1 2 + x 2 2 + x 3 2 ) = ( c t ) 2 - r 2 = - χ 2 = ( c τ ) 2 \begin{aligned}&\displaystyle\left(ct\right)^{2}-\left(x_{1}^{2}+x_{2}^{2}+x_{% 3}^{2}\right)\\ &\displaystyle=\left(ct\right)^{2}-r^{2}\\ &\displaystyle=-\chi^{2}=\left(c\tau\right)^{2}\end{aligned}\,\!
  69. 𝐩 = γ m 𝐮 \mathbf{p}=\gamma m\mathbf{u}\,\!
  70. 𝐩 𝐩 p 2 p 1 2 + p 2 2 + p 3 2 \mathbf{p}\cdot\mathbf{p}\equiv p^{2}\equiv p_{1}^{2}+p_{2}^{2}+p_{3}^{2}\,\!
  71. s y m b o l 𝖯 = m s y m b o l 𝖴 symbol{\mathsf{P}}=msymbol{\mathsf{U}}\,\!
  72. s y m b o l 𝖯 s y m b o l 𝖯 = ( m c ) 2 symbol{\mathsf{P}}\cdot symbol{\mathsf{P}}=\left(mc\right)^{2}\,\!
  73. ( E c ) 2 - ( p 1 2 + p 2 2 + p 3 2 ) = ( E c ) 2 - p 2 = ( m c ) 2 \begin{aligned}&\displaystyle\left(\frac{E}{c}\right)^{2}-\left(p_{1}^{2}+p_{2% }^{2}+p_{3}^{2}\right)\\ &\displaystyle=\left(\frac{E}{c}\right)^{2}-p^{2}\\ &\displaystyle=\left(mc\right)^{2}\end{aligned}\,\!
  74. E 2 = ( p c ) 2 + ( m c 2 ) 2 E^{2}=\left(pc\right)^{2}+\left(mc^{2}\right)^{2}\,\!
  75. 𝐮 = d 𝐫 d t \mathbf{u}=\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t}\,\!
  76. s y m b o l 𝖴 = d s y m b o l 𝖷 d τ = γ ( c , 𝐮 ) symbol{\mathsf{U}}=\frac{\mathrm{d}symbol{\mathsf{X}}}{\mathrm{d}\tau}=\gamma% \left(c,\mathbf{u}\right)
  77. s y m b o l 𝖴 s y m b o l 𝖴 = c 2 symbol{\mathsf{U}}\cdot symbol{\mathsf{U}}=c^{2}\,\!
  78. 𝐚 = d 𝐮 d t \mathbf{a}=\frac{\mathrm{d}\mathbf{u}}{\mathrm{d}t}\,\!
  79. s y m b o l 𝖠 = d s y m b o l 𝖴 d τ = γ ( c d γ d t , d γ d t 𝐮 + γ 𝐚 ) symbol{\mathsf{A}}=\frac{\mathrm{d}symbol{\mathsf{U}}}{\mathrm{d}\tau}=\gamma% \left(c\frac{\mathrm{d}\gamma}{\mathrm{d}t},\frac{\mathrm{d}\gamma}{\mathrm{d}% t}\mathbf{u}+\gamma\mathbf{a}\right)
  80. s y m b o l 𝖠 s y m b o l 𝖴 = 0 symbol{\mathsf{A}}\cdot symbol{\mathsf{U}}=0\,\!
  81. 𝐟 = d 𝐩 d t \mathbf{f}=\frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t}\,\!
  82. s y m b o l 𝖥 = d s y m b o l 𝖯 d τ = γ m ( c d γ d t , d γ d t 𝐮 + γ 𝐚 ) symbol{\mathsf{F}}=\frac{\mathrm{d}symbol{\mathsf{P}}}{\mathrm{d}\tau}=\gamma m% \left(c\frac{\mathrm{d}\gamma}{\mathrm{d}t},\frac{\mathrm{d}\gamma}{\mathrm{d}% t}\mathbf{u}+\gamma\mathbf{a}\right)
  83. s y m b o l 𝖥 s y m b o l 𝖴 = 0 symbol{\mathsf{F}}\cdot symbol{\mathsf{U}}=0\,\!
  84. ν = γ ν ( 1 - β cos θ ) \nu^{\prime}=\gamma\nu\left(1-\beta\cos\theta\right)
  85. ν = ν 1 - β 1 + β \nu^{\prime}=\nu\frac{\sqrt{1-\beta}}{\sqrt{1+\beta}}
  86. ν = γ ν \nu^{\prime}=\gamma\nu
  87. ( E c p x p y p z ) = ( γ - γ β 0 0 - γ β γ 0 0 0 0 1 0 0 0 0 1 ) ( E c p x p y p z ) \begin{pmatrix}\frac{E^{\prime}}{c}\\ p^{\prime}_{x}\\ p^{\prime}_{y}\\ p^{\prime}_{z}\end{pmatrix}=\begin{pmatrix}\gamma&-\gamma\beta&0&0\\ -\gamma\beta&\gamma&0&0\\ 0&0&1&0\\ 0&0&0&1\end{pmatrix}\begin{pmatrix}\frac{E}{c}\\ p_{x}\\ p_{y}\\ p_{z}\end{pmatrix}
  88. E c = γ E c - γ β p x \frac{E^{\prime}}{c}=\gamma\frac{E}{c}-\gamma\beta p_{x}
  89. p x = p cos θ p_{x}=\|p\|\cos\theta
  90. p \vec{p}
  91. h ν c = γ h ν c - γ β p cos θ = γ h ν c - γ β h ν c cos θ \frac{h\nu^{\prime}}{c}=\gamma\frac{h\nu}{c}-\gamma\beta\left\|p\right\|\cos% \theta=\gamma\frac{h\nu}{c}-\gamma\beta\frac{h\nu}{c}\cos\theta
  92. ν = γ ν - γ β ν cos θ = γ ν ( 1 - β cos θ ) \nu^{\prime}=\gamma\nu-\gamma\beta\nu\cos\theta=\gamma\nu\left(1-\beta\cos% \theta\right)
  93. ν \displaystyle\nu^{\prime}
  94. ν = γ ν \nu^{\prime}=\gamma\nu

List_of_Russian_people.html

  1. E = k m c 2 E=kmc^{2}

List_of_statistics_articles.html

  1. x ¯ \bar{x}
  2. x ¯ \bar{x}
  3. x ¯ \bar{x}

List_of_things_named_after_Carl_Friedrich_Gauss.html

  1. 2 \scriptstyle\sqrt{2}

List_of_unsolved_problems_in_mathematics.html

  1. γ \gamma
  2. π \pi
  3. π \pi
  4. π \pi
  5. π \pi
  6. π \pi
  7. π \pi
  8. 2 ¯ \overline{2}
  9. π \pi
  10. π \pi
  11. π \pi
  12. π \pi
  13. S n S_{n}
  14. k + 1 k+1
  15. 1 / ( k + 1 ) 1/(k+1)
  16. n n
  17. × 2 , × 3 \times 2,\times 3
  18. p c p_{c}
  19. 0 \aleph_{0}
  20. 1 \aleph_{1}
  21. p \mathbb{Z}_{p}
  22. \mathbb{C}
  23. ω 1 \aleph_{\omega_{1}}
  24. n \aleph_{n}
  25. e - 1 / 2 e^{-1/2}
  26. n > 4 n>4
  27. R ( 5 , 5 ) R(5,5)
  28. ( E c f ( λ ) λ + ) {\diamondsuit(E^{\lambda^{+}}_{cf(\lambda)}})
  29. λ \lambda

List_of_unsolved_problems_in_physics.html

  1. ν = 5 / 2 \nu=5/2
  2. ν = 8 / 5 \nu=8/5

Lists_of_integrals.html

  1. 1 x d x = ln | x | + C \int{1\over x}\,dx=\ln\left|x\right|+C
  2. π \pi
  3. π \pi
  4. 1 x d x = ln | x | + { A if x > 0 ; B if x < 0. \int{1\over x}\,dx=\ln|x|+\begin{cases}A&\,\text{if }x>0;\\ B&\,\text{if }x<0.\end{cases}
  5. k d x = k x + C \int k\,dx=kx+C
  6. x a d x = x a + 1 a + 1 + C (for a - 1 ) \int x^{a}\,dx=\frac{x^{a+1}}{a+1}+C\qquad\,\text{(for }a\neq-1\,\text{)}\,\!
  7. ( a x + b ) n d x = ( a x + b ) n + 1 a ( n + 1 ) + C (for n - 1 ) \int(ax+b)^{n}\,dx=\frac{(ax+b)^{n+1}}{a(n+1)}+C\qquad\,\text{(for }n\neq-1\,% \text{)}\,\!
  8. 1 x d x = ln | x | + C \int{1\over x}\,dx=\ln\left|x\right|+C
  9. 1 x d x = { ln | x | + C - x < 0 ln | x | + C + x > 0 \int{1\over x}\,dx=\begin{cases}\ln\left|x\right|+C^{-}&x<0\\ \ln\left|x\right|+C^{+}&x>0\end{cases}
  10. c a x + b d x = c a ln | a x + b | + C \int\frac{c}{ax+b}\,dx=\frac{c}{a}\ln\left|ax+b\right|+C
  11. e a x d x = 1 a e a x + C \int e^{ax}\,dx=\frac{1}{a}e^{ax}+C
  12. f ( x ) e f ( x ) d x = e f ( x ) + C \int f^{\prime}(x)e^{f(x)}\,dx=e^{f(x)}+C
  13. a x d x = a x ln a + C \int a^{x}\,dx=\frac{a^{x}}{\ln a}+C
  14. ln x d x = x ln x - x + C \int\ln x\,dx=x\ln x-x+C
  15. log a x d x = x log a x - x ln a + C \int\log_{a}x\,dx=x\log_{a}x-\frac{x}{\ln a}+C
  16. sin x d x = - cos x + C \int\sin{x}\,dx=-\cos{x}+C
  17. cos x d x = sin x + C \int\cos{x}\,dx=\sin{x}+C
  18. tan x d x = - ln | cos x | + C = ln | sec x | + C \int\tan{x}\,dx=-\ln{\left|\cos{x}\right|}+C=\ln{\left|\sec{x}\right|}+C
  19. cot x d x = ln | sin x | + C \int\cot{x}\,dx=\ln{\left|\sin{x}\right|}+C
  20. sec x d x = ln | sec x + tan x | + C \int\sec{x}\,dx=\ln{\left|\sec{x}+\tan{x}\right|}+C
  21. csc x d x = ln | csc x - cot x | + C \int\csc{x}\,dx=\ln{\left|\csc{x}-\cot{x}\right|}+C
  22. sec 2 x d x = tan x + C \int\sec^{2}x\,dx=\tan x+C
  23. csc 2 x d x = - cot x + C \int\csc^{2}x\,dx=-\cot x+C
  24. sec x tan x d x = sec x + C \int\sec{x}\,\tan{x}\,dx=\sec{x}+C
  25. csc x cot x d x = - csc x + C \int\csc{x}\,\cot{x}\,dx=-\csc{x}+C
  26. sin 2 x d x = 1 2 ( x - sin 2 x 2 ) + C = 1 2 ( x - sin x cos x ) + C \int\sin^{2}x\,dx=\frac{1}{2}\left(x-\frac{\sin 2x}{2}\right)+C=\frac{1}{2}(x-% \sin x\cos x)+C
  27. cos 2 x d x = 1 2 ( x + sin 2 x 2 ) + C = 1 2 ( x + sin x cos x ) + C \int\cos^{2}x\,dx=\frac{1}{2}\left(x+\frac{\sin 2x}{2}\right)+C=\frac{1}{2}(x+% \sin x\cos x)+C
  28. sec 3 x d x = 1 2 sec x tan x + 1 2 ln | sec x + tan x | + C \int\sec^{3}x\,dx=\frac{1}{2}\sec x\tan x+\frac{1}{2}\ln|\sec x+\tan x|+C
  29. sin n x d x = - sin n - 1 x cos x n + n - 1 n sin n - 2 x d x \int\sin^{n}x\,dx=-\frac{\sin^{n-1}{x}\cos{x}}{n}+\frac{n-1}{n}\int\sin^{n-2}{% x}\,dx
  30. cos n x d x = cos n - 1 x sin x n + n - 1 n cos n - 2 x d x \int\cos^{n}x\,dx=\frac{\cos^{n-1}{x}\sin{x}}{n}+\frac{n-1}{n}\int\cos^{n-2}{x% }\,dx
  31. arcsin x d x = x arcsin x + 1 - x 2 + C , for | x | + 1 \int\arcsin{x}\,dx=x\arcsin{x}+\sqrt{1-x^{2}}+C,\,\text{ for }|x|\leq+1
  32. arccos x d x = x arccos x - 1 - x 2 + C , for | x | + 1 \int\arccos{x}\,dx=x\arccos{x}-\sqrt{1-x^{2}}+C,\,\text{ for }|x|\leq+1
  33. arctan x d x = x arctan x - 1 2 ln | 1 + x 2 | + C , for all real x \int\arctan{x}\,dx=x\arctan{x}-\frac{1}{2}\ln{|1+x^{2}|}+C,\,\text{ for all % real }x
  34. \arccot x d x = x \arccot x + 1 2 ln | 1 + x 2 | + C , for all real x \int\arccot{x}\,dx=x\arccot{x}+\frac{1}{2}\ln{|1+x^{2}|}+C,\,\text{ for all % real }x
  35. \arcsec x d x = x \arcsec x - ln | x ( 1 + 1 - x - 2 ) | + C , for | x | + 1 \int\arcsec{x}\,dx=x\arcsec{x}-\ln|x\,(1+\sqrt{1-x^{-2}}\,)|+C,\,\text{ for }|% x|\geq+1
  36. \arccsc x d x = x \arccsc x + ln | x ( 1 + 1 - x - 2 ) | + C , for | x | + 1 \int\arccsc{x}\,dx=x\arccsc{x}+\ln|x\,(1+\sqrt{1-x^{-2}}\,)|+C,\,\text{ for }|% x|\geq+1
  37. sinh x d x = cosh x + C \int\sinh x\,dx=\cosh x+C
  38. cosh x d x = sinh x + C \int\cosh x\,dx=\sinh x+C
  39. tanh x d x = ln cosh x + C \int\tanh x\,dx=\ln\cosh x+C
  40. coth x d x = ln | sinh x | + C , for x 0 \int\coth x\,dx=\ln|\sinh x|+C,\,\text{ for }x\neq 0
  41. sech x d x = arctan ( sinh x ) + C \int\operatorname{sech}\,x\,dx=\arctan\,(\sinh x)+C
  42. csch x d x = ln | tanh x 2 | + C , for x 0 \int\operatorname{csch}\,x\,dx=\ln\left|\tanh{x\over 2}\right|+C,\,\text{ for % }x\neq 0
  43. arsinh x d x = x arsinh x - x 2 + 1 + C , for all real x \int\operatorname{arsinh}\,x\,dx=x\,\operatorname{arsinh}\,x-\sqrt{x^{2}+1}+C,% \,\text{ for all real }x
  44. arcosh x d x = x arcosh x - x 2 - 1 + C , for x 1 \int\operatorname{arcosh}\,x\,dx=x\,\operatorname{arcosh}\,x-\sqrt{x^{2}-1}+C,% \,\text{ for }x\geq 1
  45. artanh x d x = x artanh x + ln ( 1 - x 2 ) 2 + C , for | x | < 1 \int\operatorname{artanh}\,x\,dx=x\,\operatorname{artanh}\,x+\frac{\ln\left(\,% 1-x^{2}\right)}{2}+C,\,\text{ for }|x|<1
  46. arcoth x d x = x arcoth x + ln ( x 2 - 1 ) 2 + C , for | x | > 1 \int\operatorname{arcoth}\,x\,dx=x\,\operatorname{arcoth}\,x+\frac{\ln\left(x^% {2}-1\right)}{2}+C,\,\text{ for }|x|>1
  47. arsech x d x = x arsech x + arcsin x + C , for 0 < x 1 \int\operatorname{arsech}\,x\,dx=x\,\operatorname{arsech}\,x+\arcsin x+C,\,% \text{ for }0<x\leq 1
  48. arcsch x d x = x arcsch x + | arsinh x | + C , for x 0 \int\operatorname{arcsch}\,x\,dx=x\,\operatorname{arcsch}\,x+|\operatorname{% arsinh}\,x|+C,\,\text{ for }x\neq 0
  49. cos a x e b x d x = e b x a 2 + b 2 ( a sin a x + b cos a x ) + C \int\cos ax\,e^{bx}\,dx=\frac{e^{bx}}{a^{2}+b^{2}}\left(a\sin ax+b\cos ax% \right)+C
  50. sin a x e b x d x = e b x a 2 + b 2 ( b sin a x - a cos a x ) + C \int\sin ax\,e^{bx}\,dx=\frac{e^{bx}}{a^{2}+b^{2}}\left(b\sin ax-a\cos ax% \right)+C
  51. cos a x cosh b x d x = 1 a 2 + b 2 ( a sin a x cosh b x + b cos a x sinh b x ) + C \int\cos ax\,\cosh bx\,dx=\frac{1}{a^{2}+b^{2}}\left(a\sin ax\,\cosh bx+b\cos ax% \,\sinh bx\right)+C
  52. sin a x cosh b x d x = 1 a 2 + b 2 ( b sin a x sinh b x - a cos a x cosh b x ) + C \int\sin ax\,\cosh bx\,dx=\frac{1}{a^{2}+b^{2}}\left(b\sin ax\,\sinh bx-a\cos ax% \,\cosh bx\right)+C
  53. | f ( x ) | d x = sgn ( f ( x ) ) g ( x ) + C , \int\left|f(x)\right|\,dx=\operatorname{sgn}(f(x))g(x)+C,
  54. | ( a x + b ) n | d x = sgn ( a x + b ) ( a x + b ) n + 1 a ( n + 1 ) + C [ n is odd, and n - 1 ] . \int\left|(ax+b)^{n}\right|\,dx=\operatorname{sgn}(ax+b){(ax+b)^{n+1}\over a(n% +1)}+C\quad[\,n\,\text{ is odd, and }n\neq-1\,]\,.
  55. | tan a x | d x = - 1 a sgn ( tan a x ) ln ( | cos a x | ) + C \int\left|\tan{ax}\right|\,dx=\frac{-1}{a}\operatorname{sgn}(\tan{ax})\ln(% \left|\cos{ax}\right|)+C
  56. a x ( n π - π 2 , n π + π 2 ) ax\in\left(n\pi-\frac{\pi}{2},n\pi+\frac{\pi}{2}\right)\,
  57. | csc a x | d x = - 1 a sgn ( csc a x ) ln ( | csc a x + cot a x | ) + C \int\left|\csc{ax}\right|\,dx=\frac{-1}{a}\operatorname{sgn}(\csc{ax})\ln(% \left|\csc{ax}+\cot{ax}\right|)+C
  58. a x ( n π , n π + π ) ax\in\left(n\pi,n\pi+\pi\right)\,
  59. | sec a x | d x = 1 a sgn ( sec a x ) ln ( | sec a x + tan a x | ) + C \int\left|\sec{ax}\right|\,dx=\frac{1}{a}\operatorname{sgn}(\sec{ax})\ln(\left% |\sec{ax}+\tan{ax}\right|)+C
  60. a x ( n π - π 2 , n π + π 2 ) ax\in\left(n\pi-\frac{\pi}{2},n\pi+\frac{\pi}{2}\right)\,
  61. | cot a x | d x = 1 a sgn ( cot a x ) ln ( | sin a x | ) + C \int\left|\cot{ax}\right|\,dx=\frac{1}{a}\operatorname{sgn}(\cot{ax})\ln(\left% |\sin{ax}\right|)+C
  62. a x ( n π , n π + π ) ax\in\left(n\pi,n\pi+\pi\right)\,
  63. sgn ( f ( x ) ) f ( x ) d x \operatorname{sgn}(f(x))\int f(x)dx
  64. π \pi
  65. | sin a x | d x = 2 a a x π - 1 a cos ( a x - a x π π ) + C \int\left|\sin{ax}\right|\,dx={2\over a}\left\lfloor\frac{ax}{\pi}\right% \rfloor-{1\over a}\cos{\left(ax-\left\lfloor\frac{ax}{\pi}\right\rfloor\pi% \right)}+C\;
  66. | cos a x | d x = 2 a a x π + 1 2 + 1 a sin ( a x - a x π + 1 2 π ) + C \int\left|\cos{ax}\right|\,dx={2\over a}\left\lfloor\frac{ax}{\pi}+\frac{1}{2}% \right\rfloor+{1\over a}\sin{\left(ax-\left\lfloor\frac{ax}{\pi}+\frac{1}{2}% \right\rfloor\pi\right)}+C\;
  67. Ci ( x ) d x = x Ci ( x ) - sin x \int\operatorname{Ci}(x)\,dx=x\operatorname{Ci}(x)-\sin x
  68. Si ( x ) d x = x Si ( x ) + cos x \int\operatorname{Si}(x)\,dx=x\operatorname{Si}(x)+\cos x
  69. Ei ( x ) d x = x Ei ( x ) - e x \int\operatorname{Ei}(x)\,dx=x\operatorname{Ei}(x)-e^{x}
  70. li ( x ) d x = x li ( x ) - Ei ( 2 ln x ) \int\operatorname{li}(x)\,dx=x\operatorname{li}(x)-\operatorname{Ei}(2\ln x)
  71. li ( x ) x d x = ln x li ( x ) - x \int\frac{\operatorname{li}(x)}{x}\,dx=\ln x\,\operatorname{li}(x)-x
  72. erf ( x ) d x = e - x 2 π + x erf ( x ) \int\operatorname{erf}(x)\,dx=\frac{e^{-x^{2}}}{\sqrt{\pi}}+x\operatorname{erf% }(x)
  73. 0 x e - x d x = 1 2 π \int_{0}^{\infty}\sqrt{x}\,e^{-x}\,dx=\frac{1}{2}\sqrt{\pi}
  74. 0 e - a x 2 d x = 1 2 π a \int_{0}^{\infty}e^{-ax^{2}}\,dx=\frac{1}{2}\sqrt{\frac{\pi}{a}}
  75. a > 0 a>0
  76. 0 x 2 e - a x 2 d x = 1 4 π a 3 \int_{0}^{\infty}{x^{2}e^{-ax^{2}}\,dx}=\frac{1}{4}\sqrt{\frac{\pi}{a^{3}}}
  77. a > 0 a>0
  78. 0 x 2 n e - a x 2 d x = 2 n - 1 2 a 0 x 2 ( n - 1 ) e - a x 2 d x = ( 2 n - 1 ) ! ! 2 n + 1 π a 2 n + 1 = ( 2 n ) ! n ! 2 2 n + 1 π a 2 n + 1 \int_{0}^{\infty}x^{2n}e^{-ax^{2}}\,dx=\frac{2n-1}{2a}\int_{0}^{\infty}x^{2(n-% 1)}e^{-ax^{2}}\,dx=\frac{(2n-1)!!}{2^{n+1}}\sqrt{\frac{\pi}{a^{2n+1}}}=\frac{(% 2n)!}{n!2^{2n+1}}\sqrt{\frac{\pi}{a^{2n+1}}}
  79. a > 0 a>0
  80. n n
  81. 0 x 3 e - a x 2 d x = 1 2 a 2 \int_{0}^{\infty}{x^{3}e^{-ax^{2}}\,dx}=\frac{1}{2a^{2}}
  82. a > 0 a>0
  83. 0 x 2 n + 1 e - a x 2 d x = n a 0 x 2 n - 1 e - a x 2 d x = n ! 2 a n + 1 \int_{0}^{\infty}x^{2n+1}e^{-ax^{2}}\,dx=\frac{n}{a}\int_{0}^{\infty}x^{2n-1}e% ^{-ax^{2}}\,dx=\frac{n!}{2a^{n+1}}
  84. a > 0 a>0
  85. n = 0 , 1 , 2 , . n=0,1,2,....
  86. 0 x e x - 1 d x = π 2 6 \int_{0}^{\infty}\frac{x}{e^{x}-1}\,dx=\frac{\pi^{2}}{6}
  87. 0 x 2 e x - 1 d x = 2 ζ ( 3 ) 2.40 \int_{0}^{\infty}\frac{x^{2}}{e^{x}-1}\,dx=2\zeta(3)\simeq 2.40
  88. 0 x 3 e x - 1 d x = π 4 15 \int_{0}^{\infty}\frac{x^{3}}{e^{x}-1}\,dx=\frac{\pi^{4}}{15}
  89. 0 sin x x d x = π 2 \int_{0}^{\infty}\frac{\sin{x}}{x}\,dx=\frac{\pi}{2}
  90. 0 sin 2 x x 2 d x = π 2 \int_{0}^{\infty}\frac{\sin^{2}{x}}{x^{2}}\,dx=\frac{\pi}{2}
  91. 0 π 2 sin n x d x = 0 π 2 cos n x d x = 1 3 5 ( n - 1 ) 2 4 6 n π 2 \int_{0}^{\frac{\pi}{2}}\sin^{n}{x}\,dx=\int_{0}^{\frac{\pi}{2}}\cos^{n}{x}\,% dx=\frac{1\cdot 3\cdot 5\cdot\cdots\cdot(n-1)}{2\cdot 4\cdot 6\cdot\cdots\cdot n% }\frac{\pi}{2}
  92. n n
  93. n 2 n≥2
  94. 0 π 2 sin n x d x = 0 π 2 cos n x d x = 2 4 6 ( n - 1 ) 3 5 7 n \int_{0}^{\frac{\pi}{2}}\sin^{n}{x}\,dx=\int_{0}^{\frac{\pi}{2}}\cos^{n}{x}\,% dx=\frac{2\cdot 4\cdot 6\cdot\cdots\cdot(n-1)}{3\cdot 5\cdot 7\cdot\cdots\cdot n}
  95. n n
  96. n 3 n≥3
  97. - π π cos ( α x ) cos n ( β x ) d x = { 2 π 2 n ( n m ) | α | = | β ( 2 m - n ) | 0 otherwise \int_{-\pi}^{\pi}\cos(\alpha x)\cos^{n}(\beta x)dx=\begin{cases}\frac{2\pi}{2^% {n}}{\left({{n}\atop{m}}\right)}&|\alpha|=|\beta(2m-n)|\\ 0&\,\text{otherwise}\end{cases}
  98. α , β , m , n α,β,m,n
  99. β 0 β≠0
  100. m , n 0 m,n≥0
  101. - π π sin ( α x ) cos n ( β x ) d x = 0 \int_{-\pi}^{\pi}\sin(\alpha x)\cos^{n}(\beta x)dx=0
  102. α , β α,β
  103. n n
  104. - π π sin ( α x ) sin n ( β x ) d x = { ( - 1 ) ( n + 1 ) / 2 ( - 1 ) m 2 π 2 n ( n m ) n odd , α = β ( 2 m - n ) 0 otherwise \int_{-\pi}^{\pi}\sin(\alpha x)\sin^{n}(\beta x)dx=\begin{cases}(-1)^{(n+1)/2}% (-1)^{m}\frac{2\pi}{2^{n}}{\left({{n}\atop{m}}\right)}&n\,\text{ odd},\ \alpha% =\beta(2m-n)\\ 0&\,\text{otherwise}\end{cases}
  105. α , β , m , n α,β,m,n
  106. β 0 β≠0
  107. m , n 0 m,n≥0
  108. - π π cos ( α x ) sin n ( β x ) d x = { ( - 1 ) n / 2 ( - 1 ) m 2 π 2 n ( n m ) n even , | α | = | β ( 2 m - n ) | 0 otherwise \int_{-\pi}^{\pi}\cos(\alpha x)\sin^{n}(\beta x)dx=\begin{cases}(-1)^{n/2}(-1)% ^{m}\frac{2\pi}{2^{n}}{\left({{n}\atop{m}}\right)}&n\,\text{ even},\ |\alpha|=% |\beta(2m-n)|\\ 0&\,\text{otherwise}\end{cases}
  109. α , β , m , n α,β,m,n
  110. β 0 β≠0
  111. m , n 0 m,n≥0
  112. - e - ( a x 2 + b x + c ) d x = π a exp [ b 2 - 4 a c 4 a ] \int_{-\infty}^{\infty}e^{-(ax^{2}+bx+c)}\,dx=\sqrt{\frac{\pi}{a}}\exp\left[% \frac{b^{2}-4ac}{4a}\right]
  113. e x p u u expuu
  114. a > 0 a>0
  115. 0 x z - 1 e - x d x = Γ ( z ) \int_{0}^{\infty}x^{z-1}\,e^{-x}\,dx=\Gamma(z)
  116. Γ ( z ) \Gamma(z)
  117. 0 1 x α - 1 ( 1 - x ) β - 1 d x = Γ ( α ) Γ ( β ) Γ ( α + β ) \int_{0}^{1}x^{\alpha-1}(1-x)^{\beta-1}dx=\frac{\Gamma(\alpha)\Gamma(\beta)}{% \Gamma(\alpha+\beta)}
  118. R e ( α ) > 0 Re(α)>0
  119. R e ( β ) > 0 Re(β)>0
  120. 0 2 π e x cos θ d θ = 2 π I 0 ( x ) \int_{0}^{2\pi}e^{x\cos\theta}d\theta=2\pi I_{0}(x)
  121. 0 2 π e x cos θ + y sin θ d θ = 2 π I 0 ( x 2 + y 2 ) \int_{0}^{2\pi}e^{x\cos\theta+y\sin\theta}d\theta=2\pi I_{0}\left(\sqrt{x^{2}+% y^{2}}\right)
  122. - ( 1 + x 2 / ν ) - ( ν + 1 ) / 2 d x = ν π Γ ( ν / 2 ) Γ ( ( ν + 1 ) / 2 ) \int_{-\infty}^{\infty}(1+x^{2}/\nu)^{-(\nu+1)/2}\,dx=\frac{\sqrt{\nu\pi}\ % \Gamma(\nu/2)}{\Gamma((\nu+1)/2)}
  123. ν > 0 ν>0
  124. a b f ( x ) d x = ( b - a ) n = 1 m = 1 2 n - 1 ( - 1 ) m + 1 2 - n f ( a + m ( b - a ) 2 - n ) . \int_{a}^{b}{f(x)\,dx}=(b-a)\sum\limits_{n=1}^{\infty}{\sum\limits_{m=1}^{2^{n% }-1}{\left({-1}\right)^{m+1}}}2^{-n}f(a+m\left({b-a}\right)2^{-n}).
  125. 0 1 ln ( 1 / x ) p d x = p ! \int_{0}^{1}\ln(1/x)^{p}\,dx=p!\;
  126. 0 1 x - x d x = n = 1 n - n ( = 1.29128599706266 ) 0 1 x x d x = - n = 1 ( - n ) - n ( = 0.78343051071213 ) \begin{aligned}\displaystyle\int_{0}^{1}x^{-x}\,dx&\displaystyle=\sum_{n=1}^{% \infty}n^{-n}&&\displaystyle(=1.29128599706266\dots)\\ \displaystyle\int_{0}^{1}x^{x}\,dx&\displaystyle=-\sum_{n=1}^{\infty}(-n)^{-n}% &&\displaystyle(=0.78343051071213\dots)\end{aligned}

Lithium-ion_battery.html

  1. C 6 \mathrm{C_{6}}
  2. x x
  3. LiCoO 2 Li 1 - x CoO 2 + x Li + + x e - \mathrm{LiCoO_{2}}\leftrightarrows\mathrm{Li}_{1-x}\mathrm{CoO_{2}}+x\mathrm{% Li^{+}}+x\mathrm{e^{-}}
  4. x Li + + x e - + x C 6 x LiC 6 x\mathrm{Li^{+}}+x\mathrm{e^{-}}+x\mathrm{C_{6}}\leftrightarrows\ x\mathrm{LiC% _{6}}
  5. Li + + e - + LiCoO 2 Li 2 O + CoO \mathrm{Li^{+}}+\mathrm{e^{-}}+\mathrm{LiCoO_{2}}\rightarrow\mathrm{Li_{2}O}+% \mathrm{CoO}
  6. LiCoO 2 Li + + CoO 2 + e - \mathrm{LiCoO_{2}}\rightarrow\mathrm{Li^{+}}+\mathrm{CoO_{2}}+\mathrm{e^{-}}
  7. x {}_{x}
  8. y {}_{y}
  9. z {}_{z}
  10. 2 {}_{2}
  11. 2 {}_{2}
  12. 4 {}_{4}
  13. 4 {}_{4}
  14. 4 {}_{4}
  15. 5 {}_{5}
  16. 12 {}_{12}
  17. c t = - D ε c x \frac{\partial c}{\partial t}=-\frac{D}{\varepsilon}\frac{\partial c}{\partial x}

Lithosphere.html

  1. h 2 κ t \,h\,\sim\,2\,\sqrt{\kappa t}\,
  2. h h
  3. κ \kappa
  4. t t

Little's_law.html

  1. L = λ W L=\lambda W\,
  2. L = 10 0.5 = 5 L=10\cdot 0.5=5
  3. W = L λ = 2 10 = 0.2 W=\frac{L}{\lambda}=\frac{2}{10}=0.2

Liu_Hui.html

  1. π \pi
  2. 3.141024 < π < 3.142074 3.141024<\pi<3.142074
  3. π < 22 7 \pi<\tfrac{22}{7}
  4. 223 71 < π \tfrac{223}{71}<\pi
  5. π \pi
  6. π \pi
  7. π = 157 50 \pi=\tfrac{157}{50}
  8. π = 3.1416 \pi=3.1416
  9. π \pi

Living_polymerization.html

  1. v = [ M ] 0 - [ M ] [ I ] 0 \ v=\frac{[M]_{0}-[M]}{[I]_{0}}

Loan.html

  1. P = L c ( 1 + c ) n ( 1 + c ) n - 1 P=L\cdot\frac{c\,(1+c)^{n}}{(1+c)^{n}-1}

Local_field.html

  1. | a | := μ ( a X ) μ ( X ) |a|:=\frac{\mu(aX)}{\mu(X)}
  2. B m := { a K : | a | m } . B_{m}:=\{a\in K:|a|\leq m\}.
  3. 𝒪 = { a F : | a | 1 } \mathcal{O}=\{a\in F:|a|\leq 1\}
  4. 𝒪 × = { a F : | a | = 1 } \mathcal{O}^{\times}=\{a\in F:|a|=1\}
  5. 𝔪 \mathfrak{m}
  6. { a F : | a | < 1 } \{a\in F:|a|<1\}
  7. 𝔪 \mathfrak{m}
  8. k = 𝒪 / 𝔪 k=\mathcal{O}/\mathfrak{m}
  9. | a | = q - v ( a ) . |a|=q^{-v(a)}.
  10. v ( i = - m a i T i ) = - m v\left(\sum_{i=-m}^{\infty}a_{i}T^{i}\right)=-m
  11. U ( n ) = 1 + 𝔪 n = { u 𝒪 × : u 1 ( mod 𝔪 n ) } U^{(n)}=1+\mathfrak{m}^{n}=\left\{u\in\mathcal{O}^{\times}:u\equiv 1\,(\mathrm% {mod}\,\mathfrak{m}^{n})\right\}
  12. 𝒪 × \mathcal{O}^{\times}
  13. 𝒪 × U ( 1 ) U ( 2 ) \mathcal{O}^{\times}\supseteq U^{(1)}\supseteq U^{(2)}\supseteq\cdots
  14. 𝒪 × / U ( n ) ( 𝒪 / 𝔪 n ) × and U ( n ) / U ( n + 1 ) 𝒪 / 𝔪 \mathcal{O}^{\times}/U^{(n)}\cong\left(\mathcal{O}/\mathfrak{m}^{n}\right)^{% \times}\,\text{ and }\,U^{(n)}/U^{(n+1)}\approx\mathcal{O}/\mathfrak{m}
  15. \approx
  16. F × ( ϖ ) × μ q - 1 × U ( 1 ) F^{\times}\cong(\varpi)\times\mu_{q-1}\times U^{(1)}
  17. F × 𝐙 𝐙 / ( q - 1 ) 𝐙 p 𝐍 F^{\times}\cong\mathbf{Z}\oplus\mathbf{Z}/{(q-1)}\oplus\mathbf{Z}_{p}^{\mathbf% {N}}
  18. F × 𝐙 𝐙 / ( q - 1 ) 𝐙 / p a 𝐙 p d F^{\times}\cong\mathbf{Z}\oplus\mathbf{Z}/(q-1)\oplus\mathbf{Z}/p^{a}\oplus% \mathbf{Z}_{p}^{d}
  19. μ p a \mu_{p^{a}}

Local_homeomorphism.html

  1. f : X Y f:X\to Y\,
  2. f ( U ) f(U)
  3. f | U : U f ( U ) f|_{U}:U\to f(U)\,

Local_ring.html

  1. \mathbb{Q}
  2. f - 1 ( n ) = m f^{-1}(n)=m
  3. i = 1 m i = { 0 } \bigcap_{i=1}^{\infty}m^{i}=\{0\}
  4. ( x ) (x)
  5. e - 1 x 2 e^{-{1\over x^{2}}}
  6. m n m^{n}
  7. x n x^{n}
  8. End R ( P ) \mathrm{End}_{R}(P)
  9. M n ( R ) \mathrm{M}_{n}(R)
  10. End R ( P ) \mathrm{End}_{R}(P)

Log-normal_distribution.html

  1. 1 2 + 1 2 erf [ ln x - μ 2 σ ] \frac{1}{2}+\frac{1}{2}\,\mathrm{erf}\Big[\frac{\ln x-\mu}{\sqrt{2}\sigma}\Big]
  2. e μ + σ 2 / 2 e^{\mu+\sigma^{2}/2}
  3. e μ e^{\mu}\,
  4. e μ - σ 2 e^{\mu-\sigma^{2}}
  5. ( e σ 2 - 1 ) e 2 μ + σ 2 (e^{\sigma^{2}}\!\!-1)e^{2\mu+\sigma^{2}}
  6. ( e σ 2 + 2 ) e σ 2 - 1 (e^{\sigma^{2}}\!\!+2)\sqrt{e^{\sigma^{2}}\!\!-1}
  7. e 4 σ 2 + 2 e 3 σ 2 + 3 e 2 σ 2 - 6 e^{4\sigma^{2}}\!\!+2e^{3\sigma^{2}}\!\!+3e^{2\sigma^{2}}\!\!-6
  8. 1 2 + 1 2 ln ( 2 π σ 2 ) + μ \frac{1}{2}+\frac{1}{2}\ln(2\pi\sigma^{2})+\mu
  9. n = 0 ( i t ) n n ! e n μ + n 2 σ 2 / 2 \sum_{n=0}^{\infty}\frac{(it)^{n}}{n!}e^{n\mu+n^{2}\sigma^{2}/2}
  10. ( 1 / σ 2 0 0 2 / σ 2 ) \begin{pmatrix}1/\sigma^{2}&0\\ 0&2/\sigma^{2}\end{pmatrix}
  11. X X
  12. Y = ln ( X ) Y=\ln(X)
  13. Y Y
  14. X = exp ( Y ) X=\exp(Y)
  15. X X
  16. ln ( X ) \ln(X)
  17. X X
  18. μ \mu
  19. σ \sigma
  20. X X
  21. X X
  22. X = e μ + σ Z X=e^{\mu+\sigma Z}
  23. Z Z
  24. log a ( Y ) \log_{a}(Y)
  25. log b ( Y ) \log_{b}(Y)
  26. a , b 1 a,b\neq 1
  27. e X e^{X}
  28. a X a^{X}
  29. a a
  30. 1 \neq 1
  31. μ \mu
  32. σ \sigma
  33. m m
  34. v v
  35. μ = ln ( m 1 + v m 2 ) , σ = ln ( 1 + v m 2 ) \mu=\ln\left(\frac{m}{\sqrt{1+\frac{v}{m^{2}}}}\right),\sigma=\sqrt{\ln\left(1% +\frac{v}{m^{2}}\right)}
  36. x x
  37. x x
  38. 𝒩 ( ln x ; μ , σ ) = 1 σ 2 π exp [ - ( ln x - μ ) 2 2 σ 2 ] , x > 0. \mathcal{N}(\mbox{ln}~{}x;\mu,\sigma)=\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-% \frac{(\mbox{ln}~{}x-\mu)^{2}}{2\sigma^{2}}\right],\ \ x>0.
  39. 𝒩 ( ln x ) d ln x = 𝒩 ( ln x ) d ln x d x d x = 𝒩 ( ln x ) d x x = ln 𝒩 ( x ) d x , \mathcal{N}(\mbox{ln}~{}x)d\mbox{ln}~{}x=\mathcal{N}(\mbox{ln}~{}x)\frac{d% \mbox{ln}~{}x}{dx}dx=\mathcal{N}(\mbox{ln}~{}x)\frac{dx}{x}={\ln\mathcal{N}}(x% )dx,
  40. ln 𝒩 ( x ; μ , σ ) = 1 x σ 2 π exp [ - ( ln x - μ ) 2 2 σ 2 ] , x > 0 {\ln\mathcal{N}}(x;\mu,\sigma)=\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{(% \mbox{ln}~{}x-\mu)^{2}}{2\sigma^{2}}\right],\ \ x>0
  41. 0 x ln 𝒩 ( ξ ; μ , σ ) d ξ = 1 2 [ 1 + erf ( ln x - μ σ 2 ) ] = 1 2 erfc ( - ln x - μ σ 2 ) = Φ ( ln x - μ σ ) , \int_{0}^{x}{\ln\mathcal{N}}(\xi;\mu,\sigma)d\xi=\frac{1}{2}\left[1+% \operatorname{erf}\!\left(\frac{\ln x-\mu}{\sigma\sqrt{2}}\right)\right]=\frac% {1}{2}\operatorname{erfc}\!\left(-\frac{\ln x-\mu}{\sigma\sqrt{2}}\right)=\Phi% \bigg(\frac{\ln x-\mu}{\sigma}\bigg),
  42. E [ X n ] = e n μ + n 2 σ 2 2 \operatorname{E}[X^{n}]=\mathrm{e}^{n\mu+\frac{n^{2}\sigma^{2}}{2}}
  43. z = ln ( x ) - ( μ + n σ 2 ) σ z=\frac{\ln(x)-(\mu+n\sigma^{2})}{\sigma}
  44. E [ e t X ] \operatorname{E}[e^{tX}]
  45. t t
  46. E [ e i t X ] \operatorname{E}[e^{itX}]
  47. n = 0 ( i t ) n n ! e n μ + n 2 σ 2 / 2 \sum_{n=0}^{\infty}\frac{(it)^{n}}{n!}e^{n\mu+n^{2}\sigma^{2}/2}
  48. φ ( t ) \varphi(t)
  49. t t
  50. φ ( t ) exp ( - W 2 ( t σ 2 e μ ) + 2 W ( t σ 2 e μ ) 2 σ 2 ) 1 + W ( t σ 2 e μ ) \varphi(t)\approx\frac{\exp\bigg(-\dfrac{W^{2}(t\sigma^{2}e^{\mu})+2W(t\sigma^% {2}e^{\mu})}{2\sigma^{2}}\bigg)}{\sqrt{1+W(t\sigma^{2}e^{\mu})}}
  51. W W
  52. φ \varphi
  53. μ \mu
  54. σ \sigma
  55. GM [ X ] \mathrm{GM}[X]
  56. GSD [ X ] \mathrm{GSD}[X]
  57. E [ X ] \mathrm{E}[X]
  58. SD [ X ] \mathrm{SD}[X]
  59. GM [ X ] = e μ \mathrm{GM}[X]=e^{\mu}
  60. GSD [ X ] = e σ \mathrm{GSD}[X]=e^{\sigma}
  61. GVar [ X ] = e σ 2 \mathrm{GVar}[X]=e^{\sigma^{2}}
  62. GCV [ X ] = e σ - 1 \mathrm{GCV}[X]=e^{\sigma}-1
  63. Y = ln X Y=\ln X
  64. Med [ X ] \mathrm{Med}[X]
  65. E [ X ] \displaystyle\mathrm{E}[X]
  66. e - 1 2 σ 2 e^{-\frac{1}{2}\sigma^{2}}
  67. X X
  68. E [ X ] = e μ + 1 2 σ 2 , \displaystyle\operatorname{E}[X]=e^{\mu+\tfrac{1}{2}\sigma^{2}},
  69. μ \mu
  70. σ \sigma
  71. σ \sigma
  72. μ \displaystyle\mu
  73. s s
  74. s s
  75. X X
  76. E [ X s ] = e s μ + 1 2 s 2 σ 2 . \operatorname{E}[X^{s}]=e^{s\mu+\frac{1}{2}s^{2}\sigma^{2}}.
  77. E [ X k ] \operatorname{E}[X^{k}]
  78. k 1 k\geq 1
  79. k k
  80. ( ln f ) = 0 (\ln f)^{\prime}=0
  81. Mode [ X ] = e μ - σ 2 . \mathrm{Mode}[X]=e^{\mu-\sigma^{2}}.
  82. F X = 0.5 F_{X}=0.5
  83. Med [ X ] = e μ . \mathrm{Med}[X]=e^{\mu}\,.
  84. CV [ X ] \mathrm{CV}[X]
  85. SD [ X ] E [ X ] \frac{\mathrm{SD}[X]}{\mathrm{E}[X]}
  86. CV [ X ] = e σ 2 - 1 . \mathrm{CV}[X]=\sqrt{e^{\sigma^{2}}-1}.
  87. X X
  88. k k
  89. g ( k ) = k x ln 𝒩 ( x ) d x g(k)=\int_{k}^{\infty}\!x{\ln\mathcal{N}}(x)\,dx
  90. ln 𝒩 ( x ) {\ln\mathcal{N}}(x)
  91. X X
  92. g ( k ) = E [ X | X > k ] P ( X > k ) g(k)=\operatorname{E}[X|X>k]P(X>k)
  93. g ( k ) = k x ln 𝒩 ( x ) d x = e μ + 1 2 σ 2 Φ ( μ + σ 2 - ln k σ ) . g(k)=\int_{k}^{\infty}\!x{\ln\mathcal{N}}(x)\,dx=e^{\mu+\tfrac{1}{2}\sigma^{2}% }\,\Phi\!\left(\frac{\mu+\sigma^{2}-\ln k}{\sigma}\right).
  94. H H
  95. G G
  96. A A
  97. H = G 2 A . H=\frac{G^{2}}{A}.
  98. σ = 6 6 \sigma=\frac{\sqrt{6}}{6}
  99. σ = 1 6 \sigma=\frac{1}{\sqrt{6}}
  100. L ( x ; μ , σ ) = i = 1 n ( 1 x i ) 𝒩 ( ln x ; μ , σ ) L(x;\mu,\sigma)=\prod_{i=1}^{n}\left(\frac{1}{x}_{i}\right)\,\mathcal{N}(\ln x% ;\mu,\sigma)
  101. L L
  102. 𝒩 \mathcal{N}
  103. L ( μ , σ | x 1 , x 2 , , x n ) \displaystyle\ell_{L}(\mu,\sigma|x_{1},x_{2},\dots,x_{n})
  104. L \ell_{L}
  105. N \ell_{N}
  106. μ \mu
  107. σ \sigma
  108. μ ^ = k ln x k n , σ ^ 2 = k ( ln x k - μ ^ ) 2 n . \widehat{\mu}=\frac{\sum_{k}\ln x_{k}}{n},\widehat{\sigma}^{2}=\frac{\sum_{k}% \left(\ln x_{k}-\widehat{\mu}\right)^{2}}{n}.
  109. s y m b o l X 𝒩 ( s y m b o l μ , s y m b o l Σ ) symbolX\sim\mathcal{N}(symbol\mu,\,symbol\Sigma)
  110. s y m b o l Y = exp ( s y m b o l X ) symbolY=\exp(symbolX)
  111. E [ s y m b o l Y ] i = e μ i + 1 2 Σ i i , \operatorname{E}[symbolY]_{i}=e^{\mu_{i}+\frac{1}{2}\Sigma_{ii}},
  112. Var [ s y m b o l Y ] i j = e μ i + μ j + 1 2 ( Σ i i + Σ j j ) ( e Σ i j - 1 ) . \operatorname{Var}[symbolY]_{ij}=e^{\mu_{i}+\mu_{j}+\frac{1}{2}(\Sigma_{ii}+% \Sigma_{jj})}(e^{\Sigma_{ij}}-1).
  113. X 𝒩 ( μ , σ 2 ) X\sim\mathcal{N}(\mu,\sigma^{2})
  114. exp ( X ) ln 𝒩 ( μ , σ 2 ) . \exp(X)\sim\operatorname{\ln\mathcal{N}}(\mu,\sigma^{2}).
  115. X ln 𝒩 ( μ , σ 2 ) X\sim\operatorname{\ln\mathcal{N}}(\mu,\sigma^{2})
  116. ln ( X ) 𝒩 ( μ , σ 2 ) \ln(X)\sim\mathcal{N}(\mu,\sigma^{2})
  117. X j ln 𝒩 ( μ j , σ j 2 ) X_{j}\sim\operatorname{\ln\mathcal{N}}(\mu_{j},\sigma_{j}^{2})
  118. n n
  119. Y = j = 1 n X j Y=\textstyle\prod_{j=1}^{n}X_{j}
  120. Y Y
  121. Y ln 𝒩 ( j = 1 n μ j , j = 1 n σ j 2 ) . Y\sim\operatorname{\ln\mathcal{N}}\Big(\textstyle\sum_{j=1}^{n}\mu_{j},\ \sum_% {j=1}^{n}\sigma_{j}^{2}\Big).
  122. X j ln 𝒩 ( μ j , σ j 2 ) X_{j}\sim\operatorname{\ln\mathcal{N}}(\mu_{j},\sigma_{j}^{2})
  123. σ \sigma
  124. μ \mu
  125. Y = j = 1 n X j Y=\textstyle\sum_{j=1}^{n}X_{j}
  126. Y Y
  127. Z Z
  128. σ Z 2 \displaystyle\sigma^{2}_{Z}
  129. X j X_{j}
  130. σ j = σ \sigma_{j}=\sigma
  131. σ Z 2 \displaystyle\sigma^{2}_{Z}
  132. X ln 𝒩 ( μ , σ 2 ) X\sim\operatorname{\ln\mathcal{N}}(\mu,\sigma^{2})
  133. X + c X+c
  134. x ( c , + ) x\in(c,+\infty)
  135. E [ X + c ] = E [ X ] + c \operatorname{E}[X+c]=\operatorname{E}[X]+c
  136. Var [ X + c ] = Var [ X ] \operatorname{Var}[X+c]=\operatorname{Var}[X]
  137. X ln 𝒩 ( μ , σ 2 ) X\sim\operatorname{\ln\mathcal{N}}(\mu,\sigma^{2})
  138. a X ln 𝒩 ( μ + ln a , σ 2 ) . aX\sim\operatorname{\ln\mathcal{N}}(\mu+\ln a,\ \sigma^{2}).
  139. X ln 𝒩 ( μ , σ 2 ) X\sim\operatorname{\ln\mathcal{N}}(\mu,\sigma^{2})
  140. 1 X ln 𝒩 ( - μ , σ 2 ) . \tfrac{1}{X}\sim\operatorname{\ln\mathcal{N}}(-\mu,\ \sigma^{2}).
  141. X ln 𝒩 ( μ , σ 2 ) X\sim\operatorname{\ln\mathcal{N}}(\mu,\sigma^{2})
  142. X a ln 𝒩 ( a μ , a 2 σ 2 ) . X^{a}\sim\operatorname{\ln\mathcal{N}}(a\mu,\ a^{2}\sigma^{2}).
  143. a 0 a\neq 0\,
  144. X | Y Rayleigh ( Y ) X|Y\sim\mathrm{Rayleigh}(Y)\,
  145. Y ln 𝒩 ( μ , σ 2 ) Y\sim\operatorname{\ln\mathcal{N}}(\mu,\sigma^{2})
  146. X Suzuki ( μ , σ ) X\sim\mathrm{Suzuki}(\mu,\sigma)\,
  147. F ( x ; μ , σ ) = [ ( e μ x ) π / ( σ 3 ) + 1 ] - 1 . F(x;\mu,\sigma)=\left[\left(\frac{e^{\mu}}{x}\right)^{\pi/(\sigma\sqrt{3})}+1% \right]^{-1}.