wpmath0000001_9

Fitts's_law.html

  1. ID = log 2 ( 2 D W ) \,\text{ID}=\log_{2}\Bigg(\frac{2D}{W}\Bigg)
  2. IP = ( I D M T ) \,\text{IP}=\Bigg(\frac{ID}{MT}\Bigg)
  3. M T = a + b I D = a + b log 2 ( 2 D W ) MT=a+b\cdot ID=a+b\cdot\log_{2}\Bigg(\frac{2D}{W}\Bigg)
  4. W / 2 {W}/{2}
  5. 1 / b {1}/{b}
  6. I D = log 2 ( D W + 1 ) ID=\log_{2}\Bigg(\frac{D}{W}+1\Bigg)
  7. W e = 4.133 × S D x W_{e}=4.133\times SD_{x}
  8. I D e = log 2 ( D W e + 1 ) ID_{e}=\log_{2}\Bigg(\frac{D}{W_{e}}+1\Bigg)
  9. I P = ( I D e M T ) IP=\Bigg(\frac{ID_{e}}{MT}\Bigg)
  10. T = a + b 1 log 2 ( D ) + b 2 log 2 ( W ) T=a+b_{1}\log_{2}(D)+b_{2}\log_{2}(W)
  11. T = a + b 1 log 2 ( D + W ) + b 2 log 2 ( W ) = a + b log 2 ( D + W W k ) T=a+b_{1}\log_{2}(D+W)+b_{2}\log_{2}(W)=a+b\log_{2}\left(\frac{D+W}{W^{k}}\right)

Fixed-odds_betting.html

  1. x + 1 x+1\,
  2. x x\,
  3. - 100 x -\frac{100}{x}
  4. 100 x + 1 % \frac{100}{x+1}\%
  5. ( 100 x ) % \left(100x\right)\%
  6. + 100 x +100x\,

Floating_point.html

  1. significand × baseexponent \,\text{significand}\times\,\text{base}\text{exponent}
  2. significand , base , exponent . \,\text{significand}\in\mathbb{Z},\,\text{base}\in\mathbb{N},\,\text{exponent}% \in\mathbb{Z}.
  3. 1.2345 = 12345 significand × 10 base - 4 exponent 1.2345=\underbrace{12345}\text{significand}\times\,\underbrace{10}\text{base}% \!\!\!\!\!\!^{\overbrace{-4}\text{exponent}}
  4. s b p - 1 × b e \frac{s}{b^{\,p-1}}\times b^{e}
  5. s s
  6. p p
  7. b b
  8. e e
  9. p = 24 p=24
  10. 11001001 00001111 1101101 0 ¯ 10100010 0 11001001\ 00001111\ 1101101\underline{0}\ 10100010\ 0
  11. 11001001 00001111 1101101 1 ¯ 11001001\ 00001111\ 1101101\underline{1}
  12. s s
  13. e = 1 e=1
  14. s s
  15. ( 1 + n = 1 p - 1 bit n × 2 - n ) × 2 e \displaystyle\left(1+\sum_{n=1}^{p-1}\,\text{bit}_{n}\times 2^{-n}\right)% \times 2^{e}
  16. n n
  17. π \pi
  18. 3 \sqrt{3}
  19. sin 3 π \sin 3\pi
  20. / 1 = 0 {}^{1}/_{\infty}=0
  21. 0 × 0\times\infty
  22. ± \pm\infty
  23. 2 ( B - 1 ) ( B P - 1 ) ( U - L + 1 ) + 1 2(B-1)(B^{P-1})(U-L+1)+1
  24. B L B^{L}
  25. ( 1 - B - P ) ( B U + 1 ) (1-B^{-P})(B^{U+1})
  26. 2 y 2^{y}
  27. k 2 y - l k2^{y-l}
  28. log 2 ( x ) \log_{2}(x)
  29. c log 2 ( x + d ) c\log_{2}(x+d)
  30. 1 / x 1 - x 1/x\approx 1-x
  31. - 1 -1
  32. 2 ( e - b ) 2^{(e-b)}
  33. 2 x 2^{x}
  34. R t o t = 1 / ( 1 / R 1 + 1 / R 2 + + 1 / R n ) R_{tot}=1/(1/R_{1}+1/R_{2}+...+1/R_{n})
  35. R 1 R_{1}
  36. 1 / R 1 1/R_{1}
  37. R t o t R_{tot}
  38. × 10 - 15 \times 10^{-}15
  39. × 10 - 7 \times 10^{-}7
  40. Q ( h ) = f ( a + h ) - f ( a ) h . Q(h)=\frac{f(a+h)-f(a)}{h}.
  41. \Epsilon mach = B 1 - P , \Epsilon\text{mach}=B^{1-P},\,
  42. \Epsilon mach = 1 2 B 1 - P . \Epsilon\text{mach}=\tfrac{1}{2}B^{1-P}.
  43. | f l ( x ) - x x | \Epsilon mach . \left|\frac{fl(x)-x}{x}\right|\leq\Epsilon\text{mach}.
  44. x x
  45. y y
  46. f l ( x y ) = f l ( f l ( x 1 * y 1 ) + f l ( x 2 * y 2 ) ) fl(x\cdot y)=fl(fl(x_{1}*y_{1})+fl(x_{2}*y_{2}))
  47. f l ( ) fl()
  48. = f l ( ( x 1 * y 1 ) ( 1 + δ 1 ) + ( x 2 * y 2 ) ( 1 + δ 2 ) ) =fl((x_{1}*y_{1})(1+\delta_{1})+(x_{2}*y_{2})(1+\delta_{2}))
  49. δ n \Epsilon mach \delta_{n}\leq\Epsilon\text{mach}
  50. = ( ( x 1 * y 1 ) ( 1 + δ 1 ) + ( x 2 * y 2 ) ( 1 + δ 2 ) ) ( 1 + δ 3 ) =((x_{1}*y_{1})(1+\delta_{1})+(x_{2}*y_{2})(1+\delta_{2}))(1+\delta_{3})
  51. = ( x 1 * y 1 ) ( 1 + δ 1 ) ( 1 + δ 3 ) + ( x 2 * y 2 ) ( 1 + δ 2 ) ( 1 + δ 3 ) =(x_{1}*y_{1})(1+\delta_{1})(1+\delta_{3})+(x_{2}*y_{2})(1+\delta_{2})(1+% \delta_{3})
  52. f l ( x y ) = x ^ y ^ fl(x\cdot y)=\hat{x}\cdot\hat{y}
  53. x ^ 1 = x 1 ( 1 + δ 1 ) \hat{x}_{1}=x_{1}(1+\delta_{1})
  54. x ^ 2 = x 2 ( 1 + δ 2 ) \hat{x}_{2}=x_{2}(1+\delta_{2})
  55. y ^ 1 = y 1 ( 1 + δ 3 ) \hat{y}_{1}=y_{1}(1+\delta_{3})
  56. y ^ 2 = y 2 ( 1 + δ 3 ) \hat{y}_{2}=y_{2}(1+\delta_{3})
  57. δ n \Epsilon mach \delta_{n}\leq\Epsilon\text{mach}
  58. ( x + y ) ( x - y ) = x 2 - y 2 (x+y)(x-y)=x^{2}-y^{2}\,
  59. sin 2 θ + cos 2 θ = 1 \sin^{2}{\theta}+\cos^{2}{\theta}=1\,
  60. t 0 = 1 3 t_{0}=\frac{1}{\sqrt{3}}
  61. first form : t i + 1 = t i 2 + 1 - 1 t i second form : t i + 1 = t i t i 2 + 1 + 1 \qquad\mathrm{first\ form:}\qquad t_{i+1}=\frac{\sqrt{t_{i}^{2}+1}-1}{t_{i}}% \qquad\mathrm{second\ form:}\qquad t_{i+1}=\frac{t_{i}}{\sqrt{t_{i}^{2}+1}+1}
  62. π 6 × 2 i × t i , converging as i \pi\sim 6\times 2^{i}\times t_{i},\qquad\mathrm{converging\ as\ i\rightarrow% \infty}\,

Floor_and_ceiling_functions.html

  1. x \lfloor x\rfloor
  2. x \lceil x\rceil
  3. [ x ] [x]
  4. x \lfloor x\rfloor
  5. x \lceil x\rceil
  6. [ [ x ] ] [\![x]\!]
  7. ] ] x [ [ ]\!]x[\![
  8. { x } \{x\}
  9. { x } = x - x . \{x\}=x-\lfloor x\rfloor.
  10. 0 { x } < 1. 0\leq\{x\}<1.\;
  11. x \lfloor x\rfloor
  12. x \lceil x\rceil
  13. { x } \{x\}
  14. \mathbb{Z}
  15. x = max { m m x } , \lfloor x\rfloor=\max\,\{m\in\mathbb{Z}\mid m\leq x\},
  16. x = min { n n x } . \lceil x\rceil=\min\,\{n\in\mathbb{Z}\mid n\geq x\}.
  17. x - 1 < m x n < x + 1. x-1<m\leq x\leq n<x+1.\;
  18. x = m \lfloor x\rfloor=m\;
  19. x = n \;\lceil x\rceil=n\;
  20. x = m if and only if m x < m + 1 , x = n if and only if n - 1 < x n , x = m if and only if x - 1 < m x , x = n if and only if x n < x + 1. \begin{aligned}\displaystyle\lfloor x\rfloor=m&\displaystyle\;\;\mbox{ if and % only if }&\displaystyle m&\displaystyle\leq x<m+1,\\ \displaystyle\lceil x\rceil=n&\displaystyle\;\;\mbox{ if and only if }&% \displaystyle n-1&\displaystyle<x\leq n,\\ \displaystyle\lfloor x\rfloor=m&\displaystyle\;\;\mbox{ if and only if }&% \displaystyle x-1&\displaystyle<m\leq x,\\ \displaystyle\lceil x\rceil=n&\displaystyle\;\;\mbox{ if and only if }&% \displaystyle x&\displaystyle\leq n<x+1.\end{aligned}
  21. x < n if and only if x < n , n < x if and only if n < x , x n if and only if x n , n x if and only if n x . \begin{aligned}\displaystyle x<n&\displaystyle\;\;\mbox{ if and only if }&% \displaystyle\lfloor x\rfloor&\displaystyle<n,\\ \displaystyle n<x&\displaystyle\;\;\mbox{ if and only if }&\displaystyle n&% \displaystyle<\lceil x\rceil,\\ \displaystyle x\leq n&\displaystyle\;\;\mbox{ if and only if }&\displaystyle% \lceil x\rceil&\displaystyle\leq n,\\ \displaystyle n\leq x&\displaystyle\;\;\mbox{ if and only if }&\displaystyle n% &\displaystyle\leq\lfloor x\rfloor.\end{aligned}
  22. x + n \displaystyle\lfloor x+n\rfloor
  23. x + y \displaystyle\lfloor x\rfloor+\lfloor y\rfloor
  24. x x , \lfloor x\rfloor\leq\lceil x\rceil,
  25. x - x = { 0 if x 1 if x \lceil x\rceil-\lfloor x\rfloor=\begin{cases}0&\mbox{ if }~{}x\in\mathbb{Z}\\ 1&\mbox{ if }~{}x\not\in\mathbb{Z}\end{cases}
  26. n = n = n . \lfloor n\rfloor=\lceil n\rceil=n.
  27. x + - x \displaystyle\lfloor x\rfloor+\lceil-x\rceil
  28. x + - x = { 0 if x - 1 if x , \lfloor x\rfloor+\lfloor-x\rfloor=\begin{cases}0&\mbox{ if }~{}x\in\mathbb{Z}% \\ -1&\mbox{ if }~{}x\not\in\mathbb{Z},\end{cases}
  29. x + - x = { 0 if x 1 if x . \lceil x\rceil+\lceil-x\rceil=\begin{cases}0&\mbox{ if }~{}x\in\mathbb{Z}\\ 1&\mbox{ if }~{}x\not\in\mathbb{Z}.\end{cases}
  30. { x } + { - x } = { 0 if x 1 if x . \{x\}+\{-x\}=\begin{cases}0&\mbox{ if }~{}x\in\mathbb{Z}\\ 1&\mbox{ if }~{}x\not\in\mathbb{Z}.\end{cases}
  31. x \displaystyle\Big\lfloor\lfloor x\rfloor\Big\rfloor
  32. x \displaystyle\Big\lfloor\lceil x\rceil\Big\rfloor
  33. ( x mod y ) mod y = x mod y . (x\,\bmod\,y)\,\bmod\,y=x\,\bmod\,y.\;
  34. { x } = x mod 1. \{x\}=x\,\bmod\,1.\;
  35. 0 { m n } 1 - 1 | n | . 0\leq\left\{\frac{m}{n}\right\}\leq 1-\frac{1}{|n|}.
  36. x + m n = x + m n , \left\lfloor\frac{x+m}{n}\right\rfloor=\left\lfloor\frac{\lfloor x\rfloor+m}{n% }\right\rfloor,
  37. x + m n = x + m n . \left\lceil\frac{x+m}{n}\right\rceil=\left\lceil\frac{\lceil x\rceil+m}{n}% \right\rceil.
  38. n = n m + n - 1 m + + n - m + 1 m , n=\left\lceil\frac{n}{m}\right\rceil+\left\lceil\frac{n-1}{m}\right\rceil+% \dots+\left\lceil\frac{n-m+1}{m}\right\rceil,
  39. n = n m + n + 1 m + + n + m - 1 m . n=\left\lfloor\frac{n}{m}\right\rfloor+\left\lfloor\frac{n+1}{m}\right\rfloor+% \dots+\left\lfloor\frac{n+m-1}{m}\right\rfloor.
  40. n = n 2 + n 2 . n=\left\lfloor\frac{n}{2}\right\rfloor+\left\lceil\frac{n}{2}\right\rceil.
  41. m x = x + x - 1 m + + x - m - 1 m , \lceil mx\rceil=\left\lceil x\right\rceil+\left\lceil x-\frac{1}{m}\right% \rceil+\dots+\left\lceil x-\frac{m-1}{m}\right\rceil,
  42. m x = x + x + 1 m + + x + m - 1 m . \lfloor mx\rfloor=\left\lfloor x\right\rfloor+\left\lfloor x+\frac{1}{m}\right% \rfloor+\dots+\left\lfloor x+\frac{m-1}{m}\right\rfloor.
  43. n m = n + m - 1 m = n - 1 m + 1 , \left\lceil\frac{n}{m}\right\rceil=\left\lfloor\frac{n+m-1}{m}\right\rfloor=% \left\lfloor\frac{n-1}{m}\right\rfloor+1,
  44. n m = n - m + 1 m = n + 1 m - 1 , \left\lfloor\frac{n}{m}\right\rfloor=\left\lceil\frac{n-m+1}{m}\right\rceil=% \left\lceil\frac{n+1}{m}\right\rceil-1,
  45. i = 1 n - 1 i m n = 1 2 ( m - 1 ) ( n - 1 ) . \sum_{i=1}^{n-1}\left\lfloor\frac{im}{n}\right\rfloor=\frac{1}{2}(m-1)(n-1).
  46. m n + 2 m n + + ( n - 1 ) m n = n m + 2 n m + + ( m - 1 ) n m . \left\lfloor\frac{m}{n}\right\rfloor+\left\lfloor\frac{2m}{n}\right\rfloor+% \dots+\left\lfloor\frac{(n-1)m}{n}\right\rfloor=\left\lfloor\frac{n}{m}\right% \rfloor+\left\lfloor\frac{2n}{m}\right\rfloor+\dots+\left\lfloor\frac{(m-1)n}{% m}\right\rfloor.
  47. x n + m + x n + 2 m + x n + + ( n - 1 ) m + x n = x m + n + x m + 2 n + x m + + ( m - 1 ) n + x m . \begin{aligned}&\displaystyle\left\lfloor\frac{x}{n}\right\rfloor+\left\lfloor% \frac{m+x}{n}\right\rfloor+\left\lfloor\frac{2m+x}{n}\right\rfloor+\dots+\left% \lfloor\frac{(n-1)m+x}{n}\right\rfloor\\ \displaystyle=&\displaystyle\left\lfloor\frac{x}{m}\right\rfloor+\left\lfloor% \frac{n+x}{m}\right\rfloor+\left\lfloor\frac{2n+x}{m}\right\rfloor+\dots+\left% \lfloor\frac{(m-1)n+x}{m}\right\rfloor.\end{aligned}
  48. x / m n = x m n \left\lfloor\frac{\lfloor x/m\rfloor}{n}\right\rfloor=\left\lfloor\frac{x}{mn}\right\rfloor
  49. x / m n = x m n \left\lceil\frac{\lceil x/m\rceil}{n}\right\rceil=\left\lceil\frac{x}{mn}\right\rceil
  50. x \lfloor x\rfloor
  51. x \lceil x\rceil
  52. { x } \{x\}
  53. x mod y x\,\bmod\,y
  54. x \lfloor x\rfloor
  55. x \lceil x\rceil
  56. { x } \{x\}\;
  57. x mod y = y 2 - y π k = 1 sin ( 2 π k x y ) k for x not a multiple of y . x\,\bmod\,y=\frac{y}{2}-\frac{y}{\pi}\sum_{k=1}^{\infty}\frac{\sin\left(\frac{% 2\pi kx}{y}\right)}{k}\qquad\mbox{for }~{}x\mbox{ not a multiple of }~{}y.
  58. { x } = 1 2 - 1 π k = 1 sin ( 2 π k x ) k for x not an integer . \{x\}=\frac{1}{2}-\frac{1}{\pi}\sum_{k=1}^{\infty}\frac{\sin(2\pi kx)}{k}% \qquad\mbox{for }~{}x\mbox{ not an integer}~{}.
  59. x = x - 1 2 + 1 π k = 1 sin ( 2 π k x ) k for x not an integer . \lfloor x\rfloor=x-\frac{1}{2}+\frac{1}{\pi}\sum_{k=1}^{\infty}\frac{\sin(2\pi kx% )}{k}\qquad\mbox{for }~{}x\mbox{ not an integer}~{}.
  60. x mod y = x - y x y . x\,\bmod\,y=x-y\left\lfloor\frac{x}{y}\right\rfloor.
  61. 0 x mod y < y , 0\leq x\,\bmod\,y<y,
  62. 0 x mod y > y . 0\geq x\,\bmod\,y>y.
  63. m = p - 1 2 , n = q - 1 2 . m=\frac{p-1}{2},\;\;n=\frac{q-1}{2}.
  64. ( q p ) = ( - 1 ) q p + 2 q p + + m q p \left(\frac{q}{p}\right)=(-1)^{\left\lfloor\frac{q}{p}\right\rfloor+\left% \lfloor\frac{2q}{p}\right\rfloor+\dots+\left\lfloor\frac{mq}{p}\right\rfloor}
  65. ( p q ) = ( - 1 ) p q + 2 p q + + n p q . \left(\frac{p}{q}\right)=(-1)^{\left\lfloor\frac{p}{q}\right\rfloor+\left% \lfloor\frac{2p}{q}\right\rfloor+\dots+\left\lfloor\frac{np}{q}\right\rfloor}.
  66. q p + 2 q p + + m q p + p q + 2 p q + + n p q = m n . \left\lfloor\frac{q}{p}\right\rfloor+\left\lfloor\frac{2q}{p}\right\rfloor+% \dots+\left\lfloor\frac{mq}{p}\right\rfloor+\left\lfloor\frac{p}{q}\right% \rfloor+\left\lfloor\frac{2p}{q}\right\rfloor+\dots+\left\lfloor\frac{np}{q}% \right\rfloor=mn.
  67. ( p q ) ( q p ) = ( - 1 ) m n = ( - 1 ) p - 1 2 q - 1 2 . \left(\frac{p}{q}\right)\left(\frac{q}{p}\right)=(-1)^{mn}=(-1)^{\frac{p-1}{2}% \frac{q-1}{2}}.
  68. ( 2 p ) = ( - 1 ) p + 1 4 , \left(\frac{2}{p}\right)=(-1)^{\left\lfloor\frac{p+1}{4}\right\rfloor},
  69. ( 3 p ) = ( - 1 ) p + 1 6 . \left(\frac{3}{p}\right)=(-1)^{\left\lfloor\frac{p+1}{6}\right\rfloor}.
  70. x x
  71. x x
  72. rpi ( x ) = x + 1 2 \,\text{rpi}(x)=\left\lfloor x+\tfrac{1}{2}\right\rfloor
  73. rni ( x ) = x - 1 2 \,\text{rni}(x)=\left\lceil x-\tfrac{1}{2}\right\rceil
  74. ri ( x ) = sgn ( x ) | x | + 1 2 \,\text{ri}(x)=\operatorname{sgn}(x)\left\lfloor|x|+\tfrac{1}{2}\right\rfloor
  75. x = x + 1 2 + ( 2 x - 1 ) / 4 - ( 2 x - 1 ) / 4 - 1 \lfloor x\rceil=\left\lfloor x+\tfrac{1}{2}\right\rfloor+\left\lceil(2x-1)/4% \right\rceil-\left\lfloor(2x-1)/4\right\rfloor-1
  76. ( 2 x - 1 ) / 4 (2x-1)/4
  77. x . \lfloor x\rfloor.
  78. x \lceil x\rceil
  79. sgn ( x ) | x | \operatorname{sgn}(x)\lfloor|x|\rfloor
  80. log b k + 1 = log b ( k + 1 ) . \lfloor\log_{b}{k}\rfloor+1=\lceil\log_{b}{(k+1)}\rceil.
  81. n p + n p 2 + n p 3 + = n - k a k p - 1 \left\lfloor\frac{n}{p}\right\rfloor+\left\lfloor\frac{n}{p^{2}}\right\rfloor+% \left\lfloor\frac{n}{p^{3}}\right\rfloor+\dots=\frac{n-\sum_{k}a_{k}}{p-1}
  82. n = k a k p k n=\sum_{k}a_{k}p^{k}
  83. γ = 1 ( 1 x - 1 x ) d x , \gamma=\int_{1}^{\infty}\left({1\over\lfloor x\rfloor}-{1\over x}\right)\,dx,
  84. γ = lim n 1 n k = 1 n ( n k - n k ) , \gamma=\lim_{n\to\infty}\frac{1}{n}\,\sum_{k=1}^{n}\left(\left\lceil\frac{n}{k% }\right\rceil-\frac{n}{k}\right),
  85. γ = k = 2 ( - 1 ) k log 2 k k = 1 2 - 1 3 + 2 ( 1 4 - 1 5 + 1 6 - 1 7 ) + 3 ( 1 8 - - 1 15 ) + \gamma=\sum_{k=2}^{\infty}(-1)^{k}\frac{\left\lfloor\log_{2}k\right\rfloor}{k}% =\tfrac{1}{2}-\tfrac{1}{3}+2\left(\tfrac{1}{4}-\tfrac{1}{5}+\tfrac{1}{6}-% \tfrac{1}{7}\right)+3\left(\tfrac{1}{8}-\dots-\tfrac{1}{15}\right)+\dots
  86. a < n b ϕ ( n ) = a b ϕ ( x ) d x + a b ( { x } - 1 2 ) ϕ ( x ) d x + ( { a } - 1 2 ) ϕ ( a ) - ( { b } - 1 2 ) ϕ ( b ) . {\sum_{a<n\leq b}\phi(n)=\int_{a}^{b}\phi(x)dx+\int_{a}^{b}\left(\{x\}-\tfrac{% 1}{2}\right)\phi^{\prime}(x)dx+\left(\{a\}-\tfrac{1}{2}\right)\phi(a)-\left(\{% b\}-\tfrac{1}{2}\right)\phi(b).}
  87. ζ ( s ) = s 1 1 2 - { x } x s + 1 d x + 1 s - 1 + 1 2 . \zeta(s)=s\int_{1}^{\infty}\frac{\frac{1}{2}-\{x\}}{x^{s+1}}\;dx+\frac{1}{s-1}% +\frac{1}{2}.
  88. m = 1 n ( n m - n - 1 m ) = 2. \sum_{m=1}^{n}\left(\left\lfloor\frac{n}{m}\right\rfloor-\left\lfloor\frac{n-1% }{m}\right\rfloor\right)=2.
  89. α = m = 1 p m r - m 2 . \alpha=\sum_{m=1}^{\infty}p_{m}r^{-m^{2}}.
  90. p n = r n 2 α - r 2 n - 1 r ( n - 1 ) 2 α . p_{n}=\left\lfloor r^{n^{2}}\alpha\right\rfloor-r^{2n-1}\left\lfloor r^{(n-1)^% {2}}\alpha\right\rfloor.
  91. θ 3 , θ 9 , θ 27 , \left\lfloor\theta^{3}\right\rfloor,\left\lfloor\theta^{9}\right\rfloor,\left% \lfloor\theta^{27}\right\rfloor,\dots
  92. 2 ω , 2 2 ω , 2 2 2 ω , \left\lfloor 2^{\omega}\right\rfloor,\left\lfloor 2^{2^{\omega}}\right\rfloor,% \left\lfloor 2^{2^{2^{\omega}}}\right\rfloor,\dots
  93. π ( n ) = j = 2 n ( j - 1 ) ! + 1 j - ( j - 1 ) ! j . \pi(n)=\sum_{j=2}^{n}\left\lfloor\frac{(j-1)!+1}{j}-\left\lfloor\frac{(j-1)!}{% j}\right\rfloor\right\rfloor.
  94. π ( n ) = j = 2 n 1 k = 2 j j k k j . \pi(n)=\sum_{j=2}^{n}\left\lfloor\frac{1}{\sum_{k=2}^{j}\left\lfloor\left% \lfloor\frac{j}{k}\right\rfloor\frac{k}{j}\right\rfloor}\right\rfloor.
  95. n 3 + n + 2 6 + n + 4 6 = n 2 + n + 3 6 , \left\lfloor\tfrac{n}{3}\right\rfloor+\left\lfloor\tfrac{n+2}{6}\right\rfloor+% \left\lfloor\tfrac{n+4}{6}\right\rfloor=\left\lfloor\tfrac{n}{2}\right\rfloor+% \left\lfloor\tfrac{n+3}{6}\right\rfloor,
  96. 1 2 + n + 1 2 = 1 2 + n + 1 4 , \left\lfloor\tfrac{1}{2}+\sqrt{n+\tfrac{1}{2}}\right\rfloor=\left\lfloor\tfrac% {1}{2}+\sqrt{n+\tfrac{1}{4}}\right\rfloor,
  97. n + n + 1 = 4 n + 2 . \left\lfloor\sqrt{n}+\sqrt{n+1}\right\rfloor=\left\lfloor\sqrt{4n+2}\right\rfloor.
  98. 3 k - 2 k ( 3 2 ) k > 2 k - ( 3 2 ) k - 2 ? 3^{k}-2^{k}\left\lfloor\left(\tfrac{3}{2}\right)^{k}\right\rfloor>2^{k}-\left% \lfloor\left(\tfrac{3}{2}\right)^{k}\right\rfloor-2\;\;?
  99. π \pi
  100. f ( n ) = 10 n - 1 π - 10 10 n - 2 π f(n)=\left\lfloor 10^{n-1}\pi\right\rfloor-10\left\lfloor 10^{n-2}\pi\right\rfloor

Fluid_dynamics.html

  1. p = ρ R u T M p=\frac{\rho R_{u}T}{M}
  2. t V ρ d V = - {\partial\over\partial t}\iiint_{V}\rho\,dV=-\,{}
  3. ρ \rho
  4. ρ t + ( ρ 𝐮 ) = 0 \ {\partial\rho\over\partial t}+\nabla\cdot(\rho\mathbf{u})=0
  5. 𝐅 surf \mathbf{F}\text{surf}
  6. t V ρ 𝐮 d V = - \frac{\partial}{\partial t}\iiint_{\scriptstyle V}\rho\mathbf{u}\,dV=-\,{}
  7. ( ρ 𝐮 d 𝐒 ) 𝐮 - (\rho\mathbf{u}\cdot d\mathbf{S})\mathbf{u}-{}
  8. + V ρ 𝐟 body d V + 𝐅 surf \displaystyle{}+\iiint_{\scriptstyle V}\rho\mathbf{f}\text{body}\,dV+\mathbf{F% }\text{surf}
  9. D 𝐮 D t = 𝐅 - p ρ \ {D\mathbf{u}\over Dt}=\mathbf{F}-{\nabla p\over\rho}
  10. ρ D h D t = D p D t + ( k T ) + Φ \ \rho{Dh\over Dt}={Dp\over Dt}+\nabla\cdot\left(k\nabla T\right)+\Phi
  11. Φ \Phi
  12. D ρ D t = 0 , \frac{\mathrm{D}\rho}{\mathrm{D}t}=0\,,
  13. T - 1 T^{-1}

Fluorescence.html

  1. S 0 + h ν e x S 1 S_{0}+h\nu_{ex}\to S_{1}
  2. S 1 S 0 + h ν e m + h e a t S_{1}\to S_{0}+h\nu_{em}+heat
  3. h ν h\nu
  4. ν \nu
  5. Φ = Number of photons emitted Number of photons absorbed \Phi=\frac{\,\text{Number of photons emitted}}{\,\text{Number of photons % absorbed}}
  6. Φ = k f i k i \Phi=\frac{{k}_{f}}{\sum_{i}{k}_{i}}
  7. k f {k}_{f}
  8. i k i \sum_{i}{k}_{i}
  9. [ S 1 ] = [ S 1 ] 0 e - Γ t \left[S1\right]=\left[S1\right]_{0}e^{-\Gamma t}
  10. [ S 1 ] \left[S1\right]
  11. t t
  12. [ S 1 ] 0 \left[S1\right]_{0}
  13. Γ \Gamma
  14. Γ t o t = Γ r a d + Γ n r a d \Gamma_{tot}=\Gamma_{rad}+\Gamma_{nrad}
  15. Γ t o t \Gamma_{tot}
  16. Γ r a d \Gamma_{rad}
  17. Γ n r a d \Gamma_{nrad}
  18. r = I - I I + 2 I r={I_{\parallel}-I_{\perp}\over I_{\parallel}+2I_{\perp}}
  19. I I_{\parallel}
  20. I I_{\perp}

Flux.html

  1. j = lim A 0 I A = d I d A j=\lim\limits_{A\rightarrow 0}\frac{I}{A}=\frac{dI}{dA}
  2. I = lim Δ t 0 Δ q Δ t = d q d t I=\lim\limits_{\Delta t\rightarrow 0}\frac{\Delta q}{\Delta t}=\frac{dq}{dt}
  3. q = t 1 t 2 S 𝐣 𝐧 ^ d A d t q=\int_{t_{1}}^{t_{2}}\iint_{S}\mathbf{j}\cdot\mathbf{\hat{n}}\,{\rm d}A\,{\rm d}t
  4. 𝐧 ^ \mathbf{\hat{n}}
  5. 𝐀 = A 𝐧 ^ \mathbf{A}=A\mathbf{\hat{n}}
  6. 𝐧 ^ \mathbf{\hat{n}}
  7. 𝐣 𝐧 ^ = j cos θ \mathbf{j}\cdot\mathbf{\hat{n}}=j\cos\theta
  8. 𝐉 A = - D A B c A \mathbf{J}_{A}=-D_{AB}\nabla c_{A}
  9. σ \sigma
  10. D = 2 3 n σ k T π m D=\frac{2}{3n\sigma}\sqrt{\frac{kT}{\pi m}}
  11. ρ = ψ * ψ = | ψ | 2 . \rho=\psi^{*}\psi=|\psi|^{2}.\,
  12. d P = | ψ | 2 d 3 𝐫 . {\rm d}P=|\psi|^{2}{\rm d}^{3}\mathbf{r}.\,
  13. 𝐉 = i 2 m ( ψ ψ * - ψ * ψ ) . \mathbf{J}=\frac{i\hbar}{2m}\left(\psi\nabla\psi^{*}-\psi^{*}\nabla\psi\right).\,
  14. C 𝐄 d s y m b o l = - C 𝐁 t d 𝐬 = - d Φ D d t \oint_{C}\mathbf{E}\cdot dsymbol{\ell}=-\int_{\partial C}{\partial\mathbf{B}% \over\partial t}\cdot{\rm d}\mathbf{s}=-\frac{{\rm d}\Phi_{D}}{{\rm d}t}

Force.html

  1. F = m a \scriptstyle{\vec{F}=m\vec{a}}
  2. F = d p d t , \vec{F}=\frac{\mathrm{d}\vec{p}}{\mathrm{d}t},
  3. p \scriptstyle\vec{p}
  4. F \scriptstyle\vec{F}
  5. F = d p d t = d ( m v ) d t , \vec{F}=\frac{\mathrm{d}\vec{p}}{\mathrm{d}t}=\frac{\mathrm{d}\left(m\vec{v}% \right)}{\mathrm{d}t},
  6. v \scriptstyle\vec{v}
  7. F = m d v d t . \vec{F}=m\frac{\mathrm{d}\vec{v}}{\mathrm{d}t}.
  8. F = m a . \vec{F}=m\vec{a}.
  9. F 1 , 2 = - F 2 , 1 . \vec{F}_{1,2}=-\vec{F}_{2,1}.
  10. F 1 , 2 + F 2 , 1 = 0 \vec{F}_{1,2}+\vec{F}_{\mathrm{2,1}}=0
  11. F = 0. \sum{\vec{F}}=0.
  12. F 1 , 2 = d p 1 , 2 d t = - F 2 , 1 = - d p 2 , 1 d t \vec{F}_{1,2}=\frac{\mathrm{d}\vec{p}_{1,2}}{\mathrm{d}t}=-\vec{F}_{2,1}=-% \frac{\mathrm{d}\vec{p}_{2,1}}{\mathrm{d}t}
  13. Δ p 1 , 2 = - Δ p 2 , 1 \Delta{\vec{p}_{1,2}}=-\Delta{\vec{p}_{2,1}}
  14. Δ p = Δ p 1 , 2 + Δ p 2 , 1 = 0 \sum{\Delta{\vec{p}}}=\Delta{\vec{p}_{1,2}}+\Delta{\vec{p}_{2,1}}=0
  15. F = d p / d t \vec{F}=\mathrm{d}\vec{p}/\mathrm{d}t
  16. p = m 0 v 1 - v 2 / c 2 \vec{p}=\frac{m_{0}\vec{v}}{\sqrt{1-v^{2}/c^{2}}}
  17. v v
  18. c c
  19. m 0 m_{0}
  20. m m
  21. x x
  22. F x = γ 3 m a x F_{x}=\gamma^{3}ma_{x}\,
  23. F y = γ m a y F_{y}=\gamma ma_{y}\,
  24. F z = γ m a z F_{z}=\gamma ma_{z}\,
  25. γ = 1 1 - v 2 / c 2 . \gamma=\frac{1}{\sqrt{1-v^{2}/c^{2}}}.
  26. γ 3 m \gamma^{3}m
  27. γ m \gamma m
  28. γ \gamma
  29. v v
  30. c c
  31. γ \gamma
  32. F = m a F=ma
  33. F μ = m A μ F^{\mu}=mA^{\mu}\,
  34. F μ F^{\mu}
  35. m m
  36. A μ A^{\mu}
  37. V ( x , y , z ) V ^ ( x ^ , y ^ , z ^ ) V(x,y,z)\to{\hat{V}}(\hat{x},\hat{y},\hat{z})
  38. g \scriptstyle\vec{g}
  39. m m
  40. F = m g \vec{F}=m\vec{g}
  41. m \scriptstyle m_{\oplus}
  42. R \scriptstyle R_{\oplus}
  43. g = - G m R 2 r ^ \vec{g}=-\frac{Gm_{\oplus}}{{R_{\oplus}}^{2}}\hat{r}
  44. r ^ \scriptstyle\hat{r}
  45. G G
  46. G G
  47. G G
  48. m 1 m_{1}
  49. m 2 m_{2}
  50. F = - G m 1 m 2 r 2 r ^ \vec{F}=-\frac{Gm_{1}m_{2}}{r^{2}}\hat{r}
  51. r r
  52. r ^ \scriptstyle\hat{r}
  53. E = F q \vec{E}={\vec{F}\over{q}}
  54. q q
  55. B = F I B={F\over{I\ell}}
  56. I I
  57. \scriptstyle\ell
  58. F = q ( E + v × B ) \vec{F}=q(\vec{E}+\vec{v}\times\vec{B})
  59. F \scriptstyle\vec{F}
  60. q q
  61. E \scriptstyle\vec{E}
  62. v \scriptstyle\vec{v}
  63. B \scriptstyle\vec{B}
  64. F sf F_{\mathrm{sf}}
  65. μ sf \mu_{\mathrm{sf}}
  66. F N F_{N}
  67. 0 F sf μ sf F N . 0\leq F_{\mathrm{sf}}\leq\mu_{\mathrm{sf}}F_{\mathrm{N}}.
  68. F kf F_{\mathrm{kf}}
  69. F kf = μ kf F N , F_{\mathrm{kf}}=\mu_{\mathrm{kf}}F_{\mathrm{N}},
  70. μ kf \mu_{\mathrm{kf}}
  71. Δ x \Delta x
  72. F = - k Δ x \vec{F}=-k\Delta\vec{x}
  73. k k
  74. F V = - P \frac{\vec{F}}{V}=-\vec{\nabla}P
  75. V V
  76. P P
  77. F d = - b v \vec{F}_{\mathrm{d}}=-b\vec{v}\,
  78. b b
  79. v \scriptstyle\vec{v}
  80. σ = F A \sigma=\frac{F}{A}
  81. A A
  82. F \scriptstyle\vec{F}
  83. τ = r × F \vec{\tau}=\vec{r}\times\vec{F}
  84. r \scriptstyle\vec{r}
  85. τ = I α \vec{\tau}=I\vec{\alpha}
  86. I I
  87. α \scriptstyle\vec{\alpha}
  88. τ = d L dt , \vec{\tau}=\frac{\mathrm{d}\vec{L}}{\mathrm{dt}},
  89. L \scriptstyle\vec{L}
  90. F = - m v 2 r ^ r \vec{F}=-\frac{mv^{2}\hat{r}}{r}
  91. m m
  92. v v
  93. r r
  94. r ^ \scriptstyle\hat{r}
  95. I = t 1 t 2 F d t , \vec{I}=\int_{t_{1}}^{t_{2}}{\vec{F}\mathrm{d}t},
  96. W = x 1 x 2 F d x , W=\int_{\vec{x}_{1}}^{\vec{x}_{2}}{\vec{F}\cdot{\mathrm{d}\vec{x}}},
  97. d x \scriptstyle{d}\vec{x}
  98. d W = d W d x d x = F d x , so P = d W d t = d W d x d x d t = F v , \,\text{d}W\,=\,\frac{\,\text{d}W}{\,\text{d}\vec{x}}\,\cdot\,\,\text{d}\vec{x% }\,=\,\vec{F}\,\cdot\,\,\text{d}\vec{x},\qquad\,\text{ so }\quad P\,=\,\frac{% \,\text{d}W}{\,\text{d}t}\,=\,\frac{\,\text{d}W}{\,\text{d}\vec{x}}\,\cdot\,% \frac{\,\text{d}\vec{x}}{\,\text{d}t}\,=\,\vec{F}\,\cdot\,\vec{v},
  99. v = d x / d t {\vec{v}\,\text{ }=\,\text{ d}\vec{x}/\,\text{d}t}
  100. U ( r ) \scriptstyle{U(\vec{r})}
  101. F = - U . \vec{F}=-\vec{\nabla}U.
  102. r \scriptstyle\vec{r}
  103. F = - G m 1 m 2 r r 3 \vec{F}=-\frac{Gm_{1}m_{2}\vec{r}}{r^{3}}
  104. G G
  105. m n m_{n}
  106. F = q 1 q 2 r 4 π ϵ 0 r 3 \vec{F}=\frac{q_{1}q_{2}\vec{r}}{4\pi\epsilon_{0}r^{3}}
  107. ϵ 0 \epsilon_{0}
  108. q n q_{n}
  109. F = - k r \vec{F}=-k\vec{r}
  110. k k
  111. k g · m · s < s u p > 2 kg·m·s<sup>−2

Ford–Fulkerson_algorithm.html

  1. G ( V , E ) G(V,E)
  2. u u
  3. v v
  4. c ( u , v ) c(u,v)
  5. f ( u , v ) f(u,v)
  6. s s
  7. t t
  8. ( u , v ) E f ( u , v ) c ( u , v ) \forall(u,v)\in E\ f(u,v)\leq c(u,v)
  9. ( u , v ) E f ( u , v ) = - f ( v , u ) \forall(u,v)\in E\ f(u,v)=-f(v,u)
  10. u u
  11. v v
  12. v v
  13. u u
  14. u u
  15. s s
  16. t t
  17. ( s , u ) E f ( s , u ) = ( v , t ) E f ( v , t ) \sum_{(s,u)\in E}f(s,u)=\sum_{(v,t)\in E}f(v,t)
  18. s s
  19. t t
  20. G f ( V , E f ) G_{f}(V,E_{f})
  21. c f ( u , v ) = c ( u , v ) - f ( u , v ) c_{f}(u,v)=c(u,v)-f(u,v)
  22. v v
  23. u u
  24. f ( u , v ) > 0 f(u,v)>0
  25. c ( v , u ) = 0 c(v,u)=0
  26. c f ( v , u ) = c ( v , u ) - f ( v , u ) = f ( u , v ) > 0 c_{f}(v,u)=c(v,u)-f(v,u)=f(u,v)>0
  27. G = ( V , E ) G=(V,E)
  28. c c
  29. s s
  30. t t
  31. f f
  32. s s
  33. t t
  34. f ( u , v ) 0 f(u,v)\leftarrow 0
  35. ( u , v ) (u,v)
  36. p p
  37. s s
  38. t t
  39. G f G_{f}
  40. c f ( u , v ) > 0 c_{f}(u,v)>0
  41. ( u , v ) p (u,v)\in p
  42. c f ( p ) = min { c f ( u , v ) : ( u , v ) p } c_{f}(p)=\min\{c_{f}(u,v):(u,v)\in p\}
  43. ( u , v ) p (u,v)\in p
  44. f ( u , v ) f ( u , v ) + c f ( p ) f(u,v)\leftarrow f(u,v)+c_{f}(p)
  45. f ( v , u ) f ( v , u ) - c f ( p ) f(v,u)\leftarrow f(v,u)-c_{f}(p)
  46. G f ( V , E f ) G_{f}(V,E_{f})
  47. s s
  48. t t
  49. S S
  50. s s
  51. S S
  52. V V
  53. s s
  54. t t
  55. G ( V , E ) G(V,E)
  56. T = { t | t is a sink } T=\{t|t\,\text{ is a sink}\}
  57. S = { s | s is a source } S=\{s|s\,\text{ is a source}\}
  58. s * s^{*}
  59. ( s * , s ) (s^{*},s)
  60. s * s^{*}
  61. s S s\in S
  62. c ( s * , s ) = d s ( d s = ( s , u ) E c ( s , u ) ) c(s^{*},s)=d_{s}\;(d_{s}=\sum_{(s,u)\in E}c(s,u))
  63. t * t^{*}
  64. ( t , t * ) (t,t^{*})
  65. t T t\in T
  66. t * t^{*}
  67. c ( t , t * ) = d t ( d t = ( v , t ) E c ( v , t ) ) c(t,t^{*})=d_{t}\;(d_{t}=\sum_{(v,t)\in E}c(v,t))
  68. u u
  69. d u d_{u}
  70. u i n , u o u t u_{in},u_{out}
  71. ( u i n , u o u t ) (u_{in},u_{out})
  72. c ( u i n , u o u t ) = d u c(u_{in},u_{out})=d_{u}
  73. O ( E f ) O(Ef)
  74. E E
  75. f f
  76. O ( E ) O(E)
  77. 1 1
  78. O ( V E 2 ) O(VE^{2})
  79. A A
  80. D D
  81. 1 1
  82. A , B , C , D A,B,C,D
  83. min ( c f ( A , B ) , c f ( B , C ) , c f ( C , D ) ) = \min(c_{f}(A,B),c_{f}(B,C),c_{f}(C,D))=
  84. min ( c ( A , B ) - f ( A , B ) , c ( B , C ) - f ( B , C ) , c ( C , D ) - f ( C , D ) ) = \min(c(A,B)-f(A,B),c(B,C)-f(B,C),c(C,D)-f(C,D))=
  85. min ( 1000 - 0 , 1 - 0 , 1000 - 0 ) = 1 \min(1000-0,1-0,1000-0)=1
  86. A , C , B , D A,C,B,D
  87. min ( c f ( A , C ) , c f ( C , B ) , c f ( B , D ) ) = \min(c_{f}(A,C),c_{f}(C,B),c_{f}(B,D))=
  88. min ( c ( A , C ) - f ( A , C ) , c ( C , B ) - f ( C , B ) , c ( B , D ) - f ( B , D ) ) = \min(c(A,C)-f(A,C),c(C,B)-f(C,B),c(B,D)-f(B,D))=
  89. min ( 1000 - 0 , 0 - ( - 1 ) , 1000 - 0 ) = 1 \min(1000-0,0-(-1),1000-0)=1
  90. C C
  91. B B
  92. A , C , B , D A,C,B,D
  93. s s
  94. t t
  95. e 1 e_{1}
  96. e 2 e_{2}
  97. e 3 e_{3}
  98. 1 1
  99. r = ( 5 - 1 ) / 2 r=(\sqrt{5}-1)/2
  100. 1 1
  101. M 2 M\geq 2
  102. r r
  103. r 2 = 1 - r r^{2}=1-r
  104. p 1 = { s , v 4 , v 3 , v 2 , v 1 , t } p_{1}=\{s,v_{4},v_{3},v_{2},v_{1},t\}
  105. p 2 = { s , v 2 , v 3 , v 4 , t } p_{2}=\{s,v_{2},v_{3},v_{4},t\}
  106. p 3 = { s , v 1 , v 2 , v 3 , t } p_{3}=\{s,v_{1},v_{2},v_{3},t\}
  107. e 1 e_{1}
  108. e 2 e_{2}
  109. e 3 e_{3}
  110. r 0 = 1 r^{0}=1
  111. r r
  112. 1 1
  113. { s , v 2 , v 3 , t } \{s,v_{2},v_{3},t\}
  114. 1 1
  115. r 0 r^{0}
  116. r 1 r^{1}
  117. 0
  118. p 1 p_{1}
  119. r 1 r^{1}
  120. r 2 r^{2}
  121. 0
  122. r 1 r^{1}
  123. p 2 p_{2}
  124. r 1 r^{1}
  125. r 2 r^{2}
  126. r 1 r^{1}
  127. 0
  128. p 1 p_{1}
  129. r 2 r^{2}
  130. 0
  131. r 3 r^{3}
  132. r 2 r^{2}
  133. p 3 p_{3}
  134. r 2 r^{2}
  135. r 2 r^{2}
  136. r 3 r^{3}
  137. 0
  138. e 1 e_{1}
  139. e 2 e_{2}
  140. e 3 e_{3}
  141. r n r^{n}
  142. r n + 1 r^{n+1}
  143. 0
  144. n n\in\mathbb{N}
  145. p 1 p_{1}
  146. p 2 p_{2}
  147. p 1 p_{1}
  148. p 3 p_{3}
  149. 1 + 2 ( r 1 + r 2 ) 1+2(r^{1}+r^{2})
  150. 1 + 2 i = 1 r i = 3 + 2 r \textstyle 1+2\sum_{i=1}^{\infty}r^{i}=3+2r
  151. 2 M + 1 2M+1

Forgetting_curve.html

  1. R = e - t S R=e^{-\frac{t}{S}}
  2. R R
  3. S S
  4. t t

Formal_language.html

  1. L L
  2. L L
  3. L L
  4. L L
  5. L L
  6. L L
  7. L L
  8. L L
  9. L L
  10. L L
  11. L L
  12. L L
  13. L L
  14. L L
  15. L 1 L_{1}
  16. L 2 L_{2}
  17. L 1 L_{1}
  18. L 2 L_{2}
  19. { w | w L 1 w L 2 } \{w|w\in L_{1}\lor w\in L_{2}\}
  20. { w | w L 1 w L 2 } \{w|w\in L_{1}\land w\in L_{2}\}
  21. { w | w L 1 } \{w|w\not\in L_{1}\}
  22. L 1 L 2 = { w z | w L 1 z L 2 } L_{1}\cdot L_{2}=\{w\cdot z|w\in L_{1}\land z\in L_{2}\}
  23. L 1 * = { ϵ } { w z | w L 1 z L 1 * } L_{1}^{*}=\{\epsilon\}\cup\{w\cdot z|w\in L_{1}\land z\in L_{1}^{*}\}
  24. { w R | w L } \{w^{R}|w\in L\}
  25. { w | w L 1 w R } , R regular \{w|w\in L_{1}\land w\in R\},R\,\text{ regular}
  26. 𝒮 \mathcal{FS}
  27. 𝒮 \mathcal{FS^{\prime}}

Formal_power_series.html

  1. x 5 x^{5}
  2. x 0 x^{0}
  3. A = 1 - 3 X + 5 X 2 - 7 X 3 + 9 X 4 - 11 X 5 + . A=1-3X+5X^{2}-7X^{3}+9X^{4}-11X^{5}+\cdots.
  4. B = 2 X + 4 X 3 + 6 X 5 + , B=2X+4X^{3}+6X^{5}+\cdots,
  5. A + B = 1 - X + 5 X 2 - 3 X 3 + 9 X 4 - 5 X 5 + . A+B=1-X+5X^{2}-3X^{3}+9X^{4}-5X^{5}+\cdots.
  6. A B = 2 X - 6 X 2 + 14 X 3 - 26 X 4 + 44 X 5 + . AB=2X-6X^{2}+14X^{3}-26X^{4}+44X^{5}+\cdots.
  7. 44 X 5 = ( 1 × 6 X 5 ) + ( 5 X 2 × 4 X 3 ) + ( 9 X 4 × 2 X ) . 44X^{5}=(1\times 6X^{5})+(5X^{2}\times 4X^{3})+(9X^{4}\times 2X).
  8. 1 1 + X = 1 - X + X 2 - X 3 + X 4 - X 5 + . \frac{1}{1+X}=1-X+X^{2}-X^{3}+X^{4}-X^{5}+\cdots.
  9. [ X 3 ] ( B ) = 4 , [ X 2 ] ( X + 3 X 2 Y 3 + 10 Y 6 ) = 3 Y 3 , [ X 2 Y 3 ] ( X + 3 X 2 Y 3 + 10 Y 6 ) = 3 , [ X n ] ( 1 1 + X ) = ( - 1 ) n , [ X n ] ( X ( 1 - X ) 2 ) = n . \begin{aligned}\displaystyle\left[X^{3}\right](B)&\displaystyle=4,\\ \displaystyle\left[X^{2}\right](X+3X^{2}Y^{3}+10Y^{6})&\displaystyle=3Y^{3},\\ \displaystyle\left[X^{2}Y^{3}\right](X+3X^{2}Y^{3}+10Y^{6})&\displaystyle=3,\\ \displaystyle\left[X^{n}\right]\left(\frac{1}{1+X}\right)&\displaystyle=(-1)^{% n},\\ \displaystyle\left[X^{n}\right]\left(\frac{X}{(1-X)^{2}}\right)&\displaystyle=% n.\end{aligned}
  10. ( a n ) n 𝒩 + ( b n ) n 𝒩 = ( a n + b n ) n 𝒩 (a_{n})_{n\in\mathcal{N}}+(b_{n})_{n\in\mathcal{N}}=\left(a_{n}+b_{n}\right)_{% n\in\mathcal{N}}
  11. ( a n ) n 𝒩 × ( b n ) n 𝒩 = ( k = 0 n a k b n - k ) n 𝒩 . (a_{n})_{n\in\mathcal{N}}\times(b_{n})_{n\in\mathcal{N}}=\left(\sum_{k=0}^{n}a% _{k}b_{n-k}\right)_{n\in\mathcal{N}}.
  12. ( a 0 , a 1 , a 2 , , a n , 0 , 0 , ) = a 0 + a 1 X + + a n X n = i = 0 n a i X i ; (a_{0},a_{1},a_{2},\ldots,a_{n},0,0,\ldots)=a_{0}+a_{1}X+\cdots+a_{n}X^{n}=% \sum_{i=0}^{n}a_{i}X^{i};
  13. ( a n ) n 𝒩 (a_{n})_{n\in\mathcal{N}}
  14. i 𝒩 a i X i \textstyle\sum_{i\in\mathcal{N}}a_{i}X^{i}
  15. ( i 𝒩 a i X i ) + ( i 𝒩 b i X i ) = i 𝒩 ( a i + b i ) X i \textstyle\left(\sum_{i\in\mathcal{N}}a_{i}X^{i}\right)+\left(\sum_{i\in% \mathcal{N}}b_{i}X^{i}\right)=\sum_{i\in\mathcal{N}}(a_{i}+b_{i})X^{i}
  16. ( i 𝒩 a i X i ) × ( i 𝒩 b i X i ) = n 𝒩 ( k = 0 n a k b n - k ) X n . \textstyle\left(\sum_{i\in\mathcal{N}}a_{i}X^{i}\right)\times\left(\sum_{i\in% \mathcal{N}}b_{i}X^{i}\right)=\sum_{n\in\mathcal{N}}\left(\sum_{k=0}^{n}a_{k}b% _{n-k}\right)X^{n}.
  17. ( a 0 , a 1 , a 2 , a 3 , ) = i = 0 a i X i , ( 1 ) (a_{0},a_{1},a_{2},a_{3},\ldots)=\sum_{i=0}^{\infty}a_{i}X^{i},\qquad(1)
  18. d ( ( a n ) , ( b n ) ) = 2 - k , d((a_{n}),(b_{n}))=2^{-k},\,\!
  19. ( i 𝒩 a i X i ) × ( i 𝒩 b i X i ) = i , j 𝒩 a i b j X i + j , \textstyle\left(\sum_{i\in\mathcal{N}}a_{i}X^{i}\right)\times\left(\sum_{i\in% \mathcal{N}}b_{i}X^{i}\right)=\sum_{i,j\in\mathcal{N}}a_{i}b_{j}X^{i+j},
  20. i = 0 a i X i \textstyle\sum_{i=0}^{\infty}a_{i}X^{i}
  21. i 𝒩 X Y i \sum_{i\in\mathcal{N}}XY^{i}
  22. X 1 - Y \textstyle\frac{X}{1-Y}
  23. i 𝒩 X i Y \sum_{i\in\mathcal{N}}X^{i}Y
  24. 1 1 - X \textstyle\frac{1}{1-X}
  25. Y 1 - X \textstyle\frac{Y}{1-X}
  26. lim n ( 1 + X n ) n \lim_{n\to\infty}\left(1+\frac{X}{n}\right)^{n}
  27. exp ( X ) = n 𝒩 X n n ! \textstyle\exp(X)=\sum_{n\in\mathcal{N}}\frac{X^{n}}{n!}
  28. ( n i ) / n i {\textstyle\left({{n}\atop{i}}\right)}/n^{i}
  29. 1 i ! \textstyle\frac{1}{i!}
  30. ( k = 0 a k X k ) n = m = 0 c m X m , \left(\sum_{k=0}^{\infty}a_{k}X^{k}\right)^{n}=\sum_{m=0}^{\infty}c_{m}X^{m},
  31. c 0 = a 0 n , c m = 1 m a 0 k = 1 m ( k n - m + k ) a k c m - k , c_{0}=a_{0}^{n},\quad c_{m}=\frac{1}{ma_{0}}\sum_{k=1}^{m}(kn-m+k)a_{k}c_{m-k},
  32. A = n = 0 a n X n A=\sum_{n=0}^{\infty}a_{n}X^{n}
  33. B = b 0 + b 1 x + B=b_{0}+b_{1}x+\ldots
  34. A B A\cdot B
  35. b 0 = 1 a 0 b n = - 1 a 0 i = 1 n a i b n - i for n 1. \begin{aligned}\displaystyle b_{0}&\displaystyle=\frac{1}{a_{0}}\\ \displaystyle b_{n}&\displaystyle=-\frac{1}{a_{0}}\sum_{i=1}^{n}a_{i}b_{n-i}% \qquad\,\text{for }n\geq 1.\end{aligned}
  36. ( 1 - X ) - 1 = n = 0 X n . (1-X)^{-1}=\sum_{n=0}^{\infty}X^{n}.
  37. n = 0 b n X n n = 0 a n X n = n = 0 c n X n , \frac{\sum_{n=0}^{\infty}b_{n}X^{n}}{\sum_{n=0}^{\infty}a_{n}X^{n}}=\sum_{n=0}% ^{\infty}c_{n}X^{n},
  38. c n = 1 a 0 ( b n - k = 1 n a k c n - k ) . c_{n}=\frac{1}{a_{0}}\left(b_{n}-\sum_{k=1}^{n}a_{k}c_{n-k}\right).
  39. f ( X ) = n = 0 a n X n f(X)=\sum_{n=0}^{\infty}a_{n}X^{n}
  40. [ X m ] f ( X ) \left[X^{m}\right]f(X)
  41. [ X m ] f ( X ) = [ X m ] n = 0 a n X n = a m . \left[X^{m}\right]f(X)=\left[X^{m}\right]\sum_{n=0}^{\infty}a_{n}X^{n}=a_{m}.
  42. f ( X ) = n = 1 a n X n = a 1 X + a 2 X 2 + f(X)=\sum_{n=1}^{\infty}a_{n}X^{n}=a_{1}X+a_{2}X^{2}+\cdots
  43. g ( X ) = n = 0 b n X n = b 0 + b 1 X + b 2 X 2 + , g(X)=\sum_{n=0}^{\infty}b_{n}X^{n}=b_{0}+b_{1}X+b_{2}X^{2}+\cdots,
  44. g ( f ( X ) ) = n = 0 b n ( f ( X ) ) n = n = 0 c n X n , g(f(X))=\sum_{n=0}^{\infty}b_{n}(f(X))^{n}=\sum_{n=0}^{\infty}c_{n}X^{n},
  45. c n := k 𝒩 , | j | = n b k a j 1 a j 2 a j k . c_{n}:=\sum_{k\in\mathcal{N},|j|=n}b_{k}a_{j_{1}}a_{j_{2}}\cdots a_{j_{k}}.
  46. j 𝒩 + k j\in\mathcal{N}_{+}^{k}
  47. | j | := j 1 + + j k = n . |j|:=j_{1}+\cdots+j_{k}=n.
  48. exp ( X ) = 1 + X + X 2 2 ! + X 3 3 ! + X 4 4 ! + , \exp(X)=1+X+\frac{X^{2}}{2!}+\frac{X^{3}}{3!}+\frac{X^{4}}{4!}+\cdots,
  49. exp ( exp ( X ) - 1 ) = 1 + X + X 2 + 5 X 3 6 + 5 X 4 8 + \exp(\exp(X)-1)=1+X+X^{2}+\frac{5X^{3}}{6}+\frac{5X^{4}}{8}+\cdots
  50. exp ( exp ( X ) ) = e exp ( exp ( X ) - 1 ) = e + e X + e X 2 + 5 e X 3 6 + \exp(\exp(X))=e\exp(\exp(X)-1)=e+eX+eX^{2}+\frac{5eX^{3}}{6}+\cdots
  51. f ( X ) = k f k X k R [ [ X ] ] \textstyle f(X)=\sum_{k}f_{k}X^{k}\in R[[X]]
  52. g ( X ) = k g k X k \textstyle g(X)=\sum_{k}g_{k}X^{k}
  53. f f
  54. f f
  55. g g
  56. g g
  57. f = n 0 a n X n f=\sum_{n\geq 0}a_{n}X^{n}
  58. D f = n 1 a n n X n - 1 . Df=\sum_{n\geq 1}a_{n}nX^{n-1}.
  59. D ( a f + b g ) = a D f + b D g D(af+bg)=a\cdot Df+b\cdot Dg
  60. D ( f g ) = f ( D g ) + ( D f ) g , D(fg)=f\cdot(Dg)+(Df)\cdot g,
  61. D ( f g ) = ( D f g ) D g , D(f\circ g)=\left(Df\circ g\right)\cdot Dg,
  62. ( D k f ) ( 0 ) = k ! a k , (D^{k}f)(0)=k!a_{k},
  63. sin ( X ) := n 0 ( - 1 ) n ( 2 n + 1 ) ! X 2 n + 1 \sin(X):=\sum_{n\geq 0}\frac{(-1)^{n}}{(2n+1)!}X^{2n+1}
  64. cos ( X ) := n 0 ( - 1 ) n ( 2 n ) ! X 2 n \cos(X):=\sum_{n\geq 0}\frac{(-1)^{n}}{(2n)!}X^{2n}
  65. sin 2 ( X ) + cos 2 ( X ) = 1 , \sin^{2}(X)+\cos^{2}(X)=1,
  66. X sin ( X ) = cos ( X ) , \frac{\partial}{\partial X}\sin(X)=\cos(X),
  67. sin ( X + Y ) = sin ( X ) cos ( Y ) + cos ( X ) sin ( Y ) . \sin(X+Y)=\sin(X)\cos(Y)+\cos(X)\sin(Y).
  68. f ( X ) = n 0 a n X n . f(X)=\sum_{n\geq 0}a_{n}X^{n}.
  69. ( f + g ) ( X ) = f ( X ) + g ( X ) (f+g)(X)=f(X)+g(X)\,
  70. ( f g ) ( X ) = f ( X ) g ( X ) . (fg)(X)=f(X)g(X).\,
  71. f - 1 = n 0 a - n - 1 ( a - f ) n . f^{-1}=\sum_{n\geq 0}a^{-n-1}(a-f)^{n}.
  72. f = n \Z a n X n f=\sum_{n\in\Z}a_{n}X^{n}
  73. i \Z a i b k - i , \sum_{i\in\Z}a_{i}b_{k-i},
  74. f = D f = n \Z n a n X n - 1 f^{\prime}=Df=\sum_{n\in\Z}na_{n}X^{n-1}
  75. ord ( f ) = ord ( f ) - 1. \mathrm{ord}(f^{\prime})=\mathrm{ord}(f)-1.
  76. D : K ( ( X ) ) K ( ( X ) ) D\colon K(\!(X)\!)\to K(\!(X)\!)
  77. ker D = K \ker D=K
  78. im D = { f K ( ( X ) ) : [ X - 1 ] f = 0 } . \mathrm{im}\,D=\{f\in K((X)):[X^{-1}]f=0\}.
  79. Res : K ( ( X ) ) K \mathrm{Res}\colon K(\!(X)\!)\to K\,
  80. 0 K K ( ( X ) ) 𝐷 K ( ( X ) ) Res K 0. 0\rightarrow K\rightarrow K(\!(X)\!)\xrightarrow{D}K(\!(X)\!)\;\xrightarrow{% \mathrm{Res}}\;K\rightarrow 0.\,
  81. Res ( f ) = 0 ; \mathrm{Res}(f^{\prime})=0;\,
  82. Res ( f g ) = - Res ( f g ) ; \mathrm{Res}(fg^{\prime})=-\mathrm{Res}(f^{\prime}g);\,
  83. Res ( f / f ) = ord ( f ) , f 0 ; \mathrm{Res}(f^{\prime}/f)=\mathrm{ord}(f),\qquad\forall f\neq 0;\,
  84. Res ( ( f g ) g ) = ord ( g ) Res ( f ) if ord ( g ) > 0 ; \mathrm{Res}\left((f\circ g)g^{\prime}\right)=\mathrm{ord}(g)\mathrm{Res}(f)% \qquad\mathrm{if}\;\mathrm{ord}(g)>0;\,
  85. [ X n ] f ( X ) = Res ( X - n - 1 f ( X ) ) . [X^{n}]f(X)=\mathrm{Res}\left(X^{-n-1}f(X)\right).\,
  86. \scriptstyle\circ
  87. \scriptstyle\circ
  88. k [ X k ] g n = n [ X - n ] f - k . \textstyle k[X^{k}]g^{n}=n[X^{-n}]f^{-k}.
  89. [ X k ] g = 1 k Res ( f - k ) . [X^{k}]g=\frac{1}{k}\mathrm{Res}\left(f^{-k}\right).
  90. k [ X k ] g n = k Res ( g n X - k - 1 ) = k Res ( X n f - k - 1 f ) = - Res ( X n ( f - k ) ) = Res ( ( X n ) f - k ) = n Res ( X n - 1 f - k ) = n [ X - n ] f - k . \begin{aligned}\displaystyle k[X^{k}]g^{n}&\displaystyle=k\mathrm{Res}\left(g^% {n}X^{-k-1}\right)=k\mathrm{Res}\left(X^{n}f^{-k-1}f\,^{\prime}\right)=-% \mathrm{Res}\left(X^{n}(f^{-k})^{\prime}\right)\\ &\displaystyle=\mathrm{Res}\left(\left(X^{n}\right)^{\prime}f^{-k}\right)=n% \mathrm{Res}\left(X^{n-1}f^{-k}\right)=n[X^{-n}]f^{-k}.\end{aligned}
  91. f 1 = g 1 = 1 f_{1}=g_{1}=1
  92. 1 α [ X m ] ( f X ) α = - 1 β [ X m ] ( g X ) β \frac{1}{\alpha}[X^{m}]\left(\frac{f}{X}\right)^{\alpha}=-\frac{1}{\beta}[X^{m% }]\left(\frac{g}{X}\right)^{\beta}
  93. α c α X α \textstyle\sum_{\alpha}c_{\alpha}X^{\alpha}
  94. ( α c α X α ) + ( α d α X α ) = α ( c α + d α ) X α \left(\sum_{\alpha}c_{\alpha}X^{\alpha}\right)+\left(\sum_{\alpha}d_{\alpha}X^% {\alpha}\right)=\sum_{\alpha}(c_{\alpha}+d_{\alpha})X^{\alpha}
  95. ( α c α X α ) × ( α d α X α ) = α , β c α d β X α + β \left(\sum_{\alpha}c_{\alpha}X^{\alpha}\right)\times\left(\sum_{\alpha}d_{% \alpha}X^{\alpha}\right)=\sum_{\alpha,\beta}c_{\alpha}d_{\beta}X^{\alpha+\beta}
  96. f n = X n + X n + 1 + X n + 2 + f_{n}=X_{n}+X_{n+1}+X_{n+2}+\ldots
  97. α c α X α \textstyle\sum_{\alpha}c_{\alpha}X^{\alpha}
  98. ( α c α X α ) + ( α d α X α ) = α ( c α + d α ) X α \left(\sum_{\alpha}c_{\alpha}X^{\alpha}\right)+\left(\sum_{\alpha}d_{\alpha}X^% {\alpha}\right)=\sum_{\alpha}(c_{\alpha}+d_{\alpha})X^{\alpha}
  99. ( α c α X α ) × ( α d α X α ) = α , β c α d β X α X β \left(\sum_{\alpha}c_{\alpha}X^{\alpha}\right)\times\left(\sum_{\alpha}d_{% \alpha}X^{\alpha}\right)=\sum_{\alpha,\beta}c_{\alpha}d_{\beta}X^{\alpha}\cdot X% ^{\beta}
  100. r = w Σ * ( r , w ) w r=\sum_{w\in\Sigma^{*}}(r,w)w

Four_color_theorem.html

  1. 6 v - 2 e = 6 i = 1 D v i - i = 1 D i v i = i = 1 D ( 6 - i ) v i = 12. 6v-2e=6\sum_{i=1}^{D}v_{i}-\sum_{i=1}^{D}iv_{i}=\sum_{i=1}^{D}(6-i)v_{i}=12.
  2. i = 1 D ( 6 - i ) v i = 12. \sum_{i=1}^{D}(6-i)v_{i}=12.
  3. p = 7 + 49 - 24 χ 2 , p=\left\lfloor\frac{7+\sqrt{49-24\chi}}{2}\right\rfloor,
  4. p = 7 + 1 + 48 g 2 p=\left\lfloor\frac{7+\sqrt{1+48g}}{2}\right\rfloor

Fourier_analysis.html

  1. S ( f ) = - s ( t ) e - i 2 π f t d t . S(f)=\int_{-\infty}^{\infty}s(t)\cdot e^{-i2\pi ft}dt.
  2. s ( t ) = - S ( f ) e i 2 π f t d f , s(t)=\int_{-\infty}^{\infty}S(f)\cdot e^{i2\pi ft}df,
  3. S [ k ] = 1 P P s P ( t ) e - i 2 π k P t d t S[k]=\frac{1}{P}\int_{P}s_{P}(t)\cdot e^{-i2\pi\frac{k}{P}t}\,dt
  4. P \scriptstyle\int_{P}
  5. s P ( t ) = k = - S [ k ] e i 2 π k P t k = - + S [ k ] δ ( f - k P ) . s_{P}(t)=\sum_{k=-\infty}^{\infty}S[k]\cdot e^{i2\pi\frac{k}{P}t}\quad% \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad\sum_{k=-\infty}^{+\infty}S[k]% \ \delta\left(f-\frac{k}{P}\right).
  6. s P ( t ) = def k = - s ( t - k P ) , s_{P}(t)\ \stackrel{\,\text{def}}{=}\ \sum_{k=-\infty}^{\infty}s(t-kP),
  7. S [ k ] = 1 P S ( k P ) . S[k]=\frac{1}{P}\cdot S\left(\frac{k}{P}\right).\,
  8. S 1 / T ( f ) = def k = - S ( f - k T ) n = - s [ n ] e - i 2 π f n T Fourier series (DTFT) Poisson summation formula = { n = - s [ n ] δ ( t - n T ) } , S_{1/T}(f)\ \stackrel{\,\text{def}}{=}\ \underbrace{\sum_{k=-\infty}^{\infty}S% \left(f-\frac{k}{T}\right)\equiv\overbrace{\sum_{n=-\infty}^{\infty}s[n]\cdot e% ^{-i2\pi fnT}}^{\,\text{Fourier series (DTFT)}}}_{\,\text{Poisson summation % formula}}=\mathcal{F}\left\{\sum_{n=-\infty}^{\infty}s[n]\ \delta(t-nT)\right% \},\,
  9. s [ n ] = def T 1 / T S 1 / T ( f ) e i 2 π f n T d f = T - S ( f ) e i 2 π f n T d f = def s ( n T ) s[n]\ \stackrel{\mathrm{def}}{=}\ T\int_{1/T}S_{1/T}(f)\cdot e^{i2\pi fnT}df=T% \underbrace{\int_{-\infty}^{\infty}S(f)\cdot e^{i2\pi fnT}df}_{\stackrel{% \mathrm{def}}{=}\ s(nT)}\,
  10. S N [ k ] = 1 N T N s N [ n ] e - i 2 π k N n S k S_{N}[k]=\frac{1}{NT}\underbrace{\sum_{N}s_{N}[n]\cdot e^{-i2\pi\frac{k}{N}n}}% _{S_{k}}\,
  11. N \scriptstyle\sum_{N}
  12. s N [ n ] = 1 N N S k e i 2 π n N k , s_{N}[n]=\frac{1}{N}\sum_{N}S_{k}\cdot e^{i2\pi\frac{n}{N}k},\,
  13. N \scriptstyle\sum_{N}
  14. s N [ n ] = def k = - s [ n - k N ] , s_{N}[n]\ \stackrel{\,\text{def}}{=}\ \sum_{k=-\infty}^{\infty}s[n-kN],
  15. s [ n ] = def T s ( n T ) , s[n]\ \stackrel{\,\text{def}}{=}\ T\cdot s(nT),\,
  16. S k = S 1 / T ( k / P ) . S_{k}=S_{1/T}(k/P).\,
  17. S 1 / T ( f ) , S_{1/T}(f),\,
  18. s s\,
  19. s ( t ) s(t)\,
  20. S ( f ) = def - s ( t ) e - i 2 π f t d t S(f)\ \stackrel{\,\text{def}}{=}\ \int_{-\infty}^{\infty}s(t)\ e^{-i2\pi ft}dt\,
  21. 1 P S ( k P ) S [ k ] = def 1 P - s ( t ) e - i 2 π k P t d t 1 P P s P ( t ) e - i 2 π k P t d t \overbrace{\frac{1}{P}\cdot S\left(\frac{k}{P}\right)}^{S[k]}\ \stackrel{\,% \text{def}}{=}\ \frac{1}{P}\int_{-\infty}^{\infty}s(t)\ e^{-i2\pi\frac{k}{P}t}% \,dt\equiv\frac{1}{P}\int_{P}s_{P}(t)\ e^{-i2\pi\frac{k}{P}t}dt\,
  22. s ( t ) = - S ( f ) e i 2 π f t d f s(t)=\int_{-\infty}^{\infty}S(f)\ e^{i2\pi ft}df\,
  23. s P ( t ) = k = - S [ k ] e i 2 π k P t Poisson summation formula (Fourier series) \underbrace{s_{P}(t)=\sum_{k=-\infty}^{\infty}S[k]\cdot e^{i2\pi\frac{k}{P}t}}% _{\,\text{Poisson summation formula (Fourier series)}}\,
  24. 1 T \scriptstyle\frac{1}{T}
  25. S 1 / T ( f ) \scriptstyle S_{1/T}(f)
  26. s ( n T ) s(nT)\,
  27. 1 T S 1 / T ( f ) = def n = - s ( n T ) e - i 2 π f n T Poisson summation formula (DTFT) \underbrace{\tfrac{1}{T}\ S_{1/T}(f)\ \stackrel{\,\text{def}}{=}\ \sum_{n=-% \infty}^{\infty}s(nT)\cdot e^{-i2\pi fnT}}_{\,\text{Poisson summation formula % (DTFT)}}\,
  28. 1 T S 1 / T ( k N T ) S k = def n = - s ( n T ) e - i 2 π k n N N s P ( n T ) e - i 2 π k n N DFT \begin{aligned}\displaystyle\overbrace{\tfrac{1}{T}\ S_{1/T}\left(\frac{k}{NT}% \right)}^{S_{k}}&\displaystyle\stackrel{\,\text{def}}{=}\ \sum_{n=-\infty}^{% \infty}s(nT)\cdot e^{-i2\pi\frac{kn}{N}}\\ &\displaystyle\equiv\underbrace{\sum_{N}s_{P}(nT)\cdot e^{-i2\pi\frac{kn}{N}}}% _{\,\text{DFT}}\end{aligned}
  29. s ( n T ) = T 1 / T 1 T S 1 / T ( f ) e i 2 π f n T d f s(nT)=T\int_{1/T}\tfrac{1}{T}\ S_{1/T}(f)\cdot e^{i2\pi fnT}df\,
  30. n = - s ( n T ) δ ( t - n T ) = - 1 T S 1 / T ( f ) e i 2 π f t d f inverse Fourier transform \sum_{n=-\infty}^{\infty}s(nT)\cdot\delta(t-nT)=\underbrace{\int_{-\infty}^{% \infty}\tfrac{1}{T}\ S_{1/T}(f)\cdot e^{i2\pi ft}\,df}_{\,\text{inverse % Fourier transform}}\,
  31. s P ( n T ) = 1 N N S k e i 2 π k n N inverse DFT = 1 P N S 1 / T ( k P ) e i 2 π k n N \begin{aligned}\displaystyle s_{P}(nT)&\displaystyle=\overbrace{\tfrac{1}{N}% \sum_{N}S_{k}\cdot e^{i2\pi\frac{kn}{N}}}^{\,\text{inverse DFT}}\\ &\displaystyle=\tfrac{1}{P}\sum_{N}S_{1/T}\left(\frac{k}{P}\right)\cdot e^{i2% \pi\frac{kn}{N}}\end{aligned}
  32. x 1 , x 2 , x 3 x_{1},x_{2},x_{3}
  33. r 1 \displaystyle r_{1}
  34. P [ k = - s ( t - k P ) ] e - i 2 π k P t d t = - s ( t ) e - i 2 π k P t d t = def S ( k / P ) \int_{P}\left[\sum_{k=-\infty}^{\infty}s(t-kP)\right]\cdot e^{-i2\pi\frac{k}{P% }t}dt=\underbrace{\int_{-\infty}^{\infty}s(t)\cdot e^{-i2\pi\frac{k}{P}t}dt}_{% \stackrel{\mathrm{def}}{=}\ S(k/P)}
  35. n = - + T s ( n T ) δ ( t - n T ) = n = - + T s ( t ) δ ( t - n T ) = s ( t ) T n = - + δ ( t - n T ) . \scriptstyle\sum_{n=-\infty}^{+\infty}T\ s(nT)\ \delta(t-nT)\ =\ \sum_{n=-% \infty}^{+\infty}T\ s(t)\ \delta(t-nT)\ =\ s(t)\cdot T\sum_{n=-\infty}^{+% \infty}\delta(t-nT).

Fourier_series.html

  1. s N ( x ) = A 0 2 + n = 1 N A n sin ( 2 π n x P + ϕ n ) , for integer N 1. s_{N}(x)=\frac{A_{0}}{2}+\sum_{n=1}^{N}A_{n}\cdot\sin(\tfrac{2\pi nx}{P}+\phi_% {n}),\quad\scriptstyle\,\text{for integer}\ N\ \geq\ 1.
  2. s N ( x ) s_{N}(x)
  3. sin ( 2 π n x P + ϕ n ) sin ( ϕ n ) cos ( 2 π n x P ) + cos ( ϕ n ) sin ( 2 π n x P ) \sin(\tfrac{2\pi nx}{P}+\phi_{n})\equiv\sin(\phi_{n})\cos(\tfrac{2\pi nx}{P})+% \cos(\phi_{n})\sin(\tfrac{2\pi nx}{P})
  4. sin ( 2 π n x P + ϕ n ) Re { 1 i e i ( 2 π n x P + ϕ n ) } = 1 2 i e i ( 2 π n x P + ϕ n ) + ( 1 2 i e i ( 2 π n x P + ϕ n ) ) * , \sin(\tfrac{2\pi nx}{P}+\phi_{n})\equiv\,\text{Re}\left\{\frac{1}{i}\cdot e^{i% \left(\tfrac{2\pi nx}{P}+\phi_{n}\right)}\right\}=\frac{1}{2i}\cdot e^{i\left(% \tfrac{2\pi nx}{P}+\phi_{n}\right)}+\left(\frac{1}{2i}\cdot e^{i\left(\tfrac{2% \pi nx}{P}+\phi_{n}\right)}\right)^{*},
  5. s N ( x ) = a 0 2 + n = 1 N ( a n A n sin ( ϕ n ) cos ( 2 π n x P ) + b n A n cos ( ϕ n ) sin ( 2 π n x P ) ) = n = - N N c n e i 2 π n x P , \begin{aligned}\displaystyle s_{N}(x)&\displaystyle=\frac{a_{0}}{2}+\sum_{n=1}% ^{N}\left(\overbrace{a_{n}}^{A_{n}\sin(\phi_{n})}\cos(\tfrac{2\pi nx}{P})+% \overbrace{b_{n}}^{A_{n}\cos(\phi_{n})}\sin(\tfrac{2\pi nx}{P})\right)\\ &\displaystyle=\sum_{n=-N}^{N}c_{n}\cdot e^{i\tfrac{2\pi nx}{P}},\end{aligned}
  6. A n A_{n}
  7. ϕ n \phi_{n}
  8. A n = a n 2 + b n 2 \displaystyle A_{n}=\sqrt{{a_{n}}^{2}+{b_{n}}^{2}}
  9. c n = def { A n 2 i e i ϕ n = 1 2 ( a n - i b n ) for n > 0 1 2 a 0 for n = 0 c | n | * for n < 0. c_{n}\ \stackrel{\mathrm{def}}{=}\ \begin{cases}\frac{A_{n}}{2i}e^{i\phi_{n}}=% \frac{1}{2}(a_{n}-ib_{n})&\,\text{for }n>0\\ \frac{1}{2}a_{0}&\,\text{for }n=0\\ c_{|n|}^{*}&\,\text{for }n<0.\end{cases}
  10. a n = 2 P x 0 x 0 + P s ( x ) cos ( 2 π n x P ) d x a_{n}=\frac{2}{P}\int_{x_{0}}^{x_{0}+P}s(x)\cdot\cos(\tfrac{2\pi nx}{P})\ dx
  11. b n = 2 P x 0 x 0 + P s ( x ) sin ( 2 π n x P ) d x b_{n}=\frac{2}{P}\int_{x_{0}}^{x_{0}+P}s(x)\cdot\sin(\tfrac{2\pi nx}{P})\ dx
  12. c n = 1 P x 0 x 0 + P s ( x ) e - i 2 π n x P d x , c_{n}=\frac{1}{P}\int_{x_{0}}^{x_{0}+P}s(x)\cdot e^{-i\tfrac{2\pi nx}{P}}\ dx,
  13. s N ( x ) s_{N}(x)
  14. s ( x ) \scriptstyle s(x)
  15. [ x 0 , x 0 + P ] , \scriptstyle[x_{0},\ x_{0}+P],
  16. s ( x ) , \scriptstyle s_{\infty}(x),
  17. s . s.
  18. s ( x ) = x π , for - π < x < π , s(x)=\frac{x}{\pi},\quad\mathrm{for}-\pi<x<\pi,
  19. s ( x + 2 π k ) = s ( x ) , for - < x < and k . s(x+2\pi k)=s(x),\quad\mathrm{for}-\infty<x<\infty\,\text{ and }k\in\mathbb{Z}.
  20. a n = 1 π - π π s ( x ) cos ( n x ) d x = 0 , n 0. b n = 1 π - π π s ( x ) sin ( n x ) d x = - 2 π n cos ( n π ) + 2 π 2 n 2 sin ( n π ) = 2 ( - 1 ) n + 1 π n , n 1. \begin{aligned}\displaystyle a_{n}&\displaystyle{}=\frac{1}{\pi}\int_{-\pi}^{% \pi}s(x)\cos(nx)\,dx=0,\quad n\geq 0.\\ \displaystyle b_{n}&\displaystyle{}=\frac{1}{\pi}\int_{-\pi}^{\pi}s(x)\sin(nx)% \,dx\\ &\displaystyle=-\frac{2}{\pi n}\cos(n\pi)+\frac{2}{\pi^{2}n^{2}}\sin(n\pi)\\ &\displaystyle=\frac{2\,(-1)^{n+1}}{\pi n},\quad n\geq 1.\end{aligned}
  21. T ( x , y ) = 2 n = 1 ( - 1 ) n + 1 n sin ( n x ) sinh ( n y ) sinh ( n π ) . T(x,y)=2\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}\sin(nx){\sinh(ny)\over\sinh(n% \pi)}.
  22. s ^ \scriptstyle\hat{s}
  23. s ( x ) \displaystyle s_{\infty}(x)
  24. S ( f ) = def n = - S [ n ] δ ( f - n P ) , S(f)\ \stackrel{\mathrm{def}}{=}\ \sum_{n=-\infty}^{\infty}S[n]\cdot\delta% \left(f-\frac{n}{P}\right),
  25. s ( x ) \scriptstyle s_{\infty}(x)
  26. - 1 { S ( f ) } \displaystyle\mathcal{F}^{-1}\{S(f)\}
  27. a k = - 1 1 φ ( y ) cos ( 2 k + 1 ) π y 2 d y = - 1 1 ( a cos π y 2 cos ( 2 k + 1 ) π y 2 + a cos 3 π y 2 cos ( 2 k + 1 ) π y 2 + ) d y \begin{aligned}\displaystyle a_{k}&\displaystyle=\int_{-1}^{1}\varphi(y)\cos(2% k+1)\frac{\pi y}{2}\,dy\\ &\displaystyle=\int_{-1}^{1}\left(a\cos\frac{\pi y}{2}\cos(2k+1)\frac{\pi y}{2% }+a^{\prime}\cos 3\frac{\pi y}{2}\cos(2k+1)\frac{\pi y}{2}+\cdots\right)\,dy% \end{aligned}
  28. cos ( 2 j + 1 ) π y 2 cos ( 2 k + 1 ) π y 2 \cos(2j+1)\frac{\pi y}{2}\cos(2k+1)\frac{\pi y}{2}
  29. f ( x , y ) = j , k 𝐙 (integers) c j , k e i j x e i k y , f(x,y)=\sum_{j,k\in\mathbf{Z}\,\text{ (integers)}}c_{j,k}e^{ijx}e^{iky},
  30. c j , k = 1 4 π 2 - π π - π π f ( x , y ) e - i j x e - i k y d x d y . c_{j,k}={1\over 4\pi^{2}}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}f(x,y)e^{-ijx}e^{-% iky}\,dx\,dy.
  31. 𝐑 = n 1 𝐚 1 + n 2 𝐚 2 + n 3 𝐚 3 \mathbf{R}=n_{1}\mathbf{a}_{1}+n_{2}\mathbf{a}_{2}+n_{3}\mathbf{a}_{3}
  32. 𝐫 = x 1 𝐚 1 a 1 + x 2 𝐚 2 a 2 + x 3 𝐚 3 a 3 , \mathbf{r}=x_{1}\frac{\mathbf{a}_{1}}{a_{1}}+x_{2}\frac{\mathbf{a}_{2}}{a_{2}}% +x_{3}\frac{\mathbf{a}_{3}}{a_{3}},
  33. g ( x 1 , x 2 , x 3 ) := f ( 𝐫 ) = f ( x 1 𝐚 1 a 1 + x 2 𝐚 2 a 2 + x 3 𝐚 3 a 3 ) . g(x_{1},x_{2},x_{3}):=f(\mathbf{r})=f\left(x_{1}\frac{\mathbf{a}_{1}}{a_{1}}+x% _{2}\frac{\mathbf{a}_{2}}{a_{2}}+x_{3}\frac{\mathbf{a}_{3}}{a_{3}}\right).
  34. g ( x 1 , x 2 , x 3 ) g(x_{1},x_{2},x_{3})
  35. g ( x 1 , x 2 , x 3 ) = g ( x 1 + a 1 , x 2 , x 3 ) = g ( x 1 , x 2 + a 2 , x 3 ) = g ( x 1 , x 2 , x 3 + a 3 ) g(x_{1},x_{2},x_{3})=g(x_{1}+a_{1},x_{2},x_{3})=g(x_{1},x_{2}+a_{2},x_{3})=g(x% _{1},x_{2},x_{3}+a_{3})
  36. h one ( m 1 , x 2 , x 3 ) := 1 a 1 0 a 1 g ( x 1 , x 2 , x 3 ) e - i 2 π m 1 a 1 x 1 d x 1 h^{\mathrm{one}}(m_{1},x_{2},x_{3}):=\frac{1}{a_{1}}\int_{0}^{a_{1}}g(x_{1},x_% {2},x_{3})\cdot e^{-i2\pi\frac{m_{1}}{a_{1}}x_{1}}\,dx_{1}
  37. g ( x 1 , x 2 , x 3 ) = m 1 = - h one ( m 1 , x 2 , x 3 ) e i 2 π m 1 a 1 x 1 g(x_{1},x_{2},x_{3})=\sum_{m_{1}=-\infty}^{\infty}h^{\mathrm{one}}(m_{1},x_{2}% ,x_{3})\cdot e^{i2\pi\frac{m_{1}}{a_{1}}x_{1}}
  38. h two ( m 1 , m 2 , x 3 ) \displaystyle h^{\mathrm{two}}(m_{1},m_{2},x_{3})
  39. g ( x 1 , x 2 , x 3 ) = m 1 = - m 2 = - h two ( m 1 , m 2 , x 3 ) e i 2 π m 1 a 1 x 1 e i 2 π m 2 a 2 x 2 g(x_{1},x_{2},x_{3})=\sum_{m_{1}=-\infty}^{\infty}\sum_{m_{2}=-\infty}^{\infty% }h^{\mathrm{two}}(m_{1},m_{2},x_{3})\cdot e^{i2\pi\frac{m_{1}}{a_{1}}x_{1}}% \cdot e^{i2\pi\frac{m_{2}}{a_{2}}x_{2}}
  40. h three ( m 1 , m 2 , m 3 ) \displaystyle h^{\mathrm{three}}(m_{1},m_{2},m_{3})
  41. g ( x 1 , x 2 , x 3 ) = m 1 = - m 2 = - m 3 = - h three ( m 1 , m 2 , m 3 ) e i 2 π m 1 a 1 x 1 e i 2 π m 2 a 2 x 2 e i 2 π m 3 a 3 x 3 g(x_{1},x_{2},x_{3})=\sum_{m_{1}=-\infty}^{\infty}\sum_{m_{2}=-\infty}^{\infty% }\sum_{m_{3}=-\infty}^{\infty}h^{\mathrm{three}}(m_{1},m_{2},m_{3})\cdot e^{i2% \pi\frac{m_{1}}{a_{1}}x_{1}}\cdot e^{i2\pi\frac{m_{2}}{a_{2}}x_{2}}\cdot e^{i2% \pi\frac{m_{3}}{a_{3}}x_{3}}
  42. g ( x 1 , x 2 , x 3 ) = m 1 , m 2 , m 3 \Z h three ( m 1 , m 2 , m 3 ) e i 2 π ( m 1 a 1 x 1 + m 2 a 2 x 2 + m 3 a 3 x 3 ) . g(x_{1},x_{2},x_{3})=\sum_{m_{1},m_{2},m_{3}\in\Z}h^{\mathrm{three}}(m_{1},m_{% 2},m_{3})\cdot e^{i2\pi\left(\frac{m_{1}}{a_{1}}x_{1}+\frac{m_{2}}{a_{2}}x_{2}% +\frac{m_{3}}{a_{3}}x_{3}\right)}.
  43. 𝐊 = l 1 𝐠 1 + l 2 𝐠 2 + l 3 𝐠 3 \mathbf{K}=l_{1}\mathbf{g}_{1}+l_{2}\mathbf{g}_{2}+l_{3}\mathbf{g}_{3}
  44. 𝐠 𝐢 𝐚 𝐣 = 2 π δ i j \mathbf{g_{i}}\cdot\mathbf{a_{j}}=2\pi\delta_{ij}
  45. 𝐊 𝐫 = ( l 1 𝐠 1 + l 2 𝐠 2 + l 3 𝐠 3 ) ( x 1 𝐚 1 a 1 + x 2 𝐚 2 a 2 + x 3 𝐚 3 a 3 ) = 2 π ( x 1 l 1 a 1 + x 2 l 2 a 2 + x 3 l 3 a 3 ) . \mathbf{K}\cdot\mathbf{r}=\left(l_{1}\mathbf{g}_{1}+l_{2}\mathbf{g}_{2}+l_{3}% \mathbf{g}_{3}\right)\cdot\left(x_{1}\frac{\mathbf{a}_{1}}{a_{1}}+x_{2}\frac{% \mathbf{a}_{2}}{a_{2}}+x_{3}\frac{\mathbf{a}_{3}}{a_{3}}\right)=2\pi\left(x_{1% }\frac{l_{1}}{a_{1}}+x_{2}\frac{l_{2}}{a_{2}}+x_{3}\frac{l_{3}}{a_{3}}\right).
  46. f ( 𝐫 ) = 𝐊 h ( 𝐊 ) e i 𝐊 𝐫 , f(\mathbf{r})=\sum_{\mathbf{K}}h(\mathbf{K})\cdot e^{i\mathbf{K}\cdot\mathbf{r% }},
  47. h ( 𝐊 ) = 1 a 3 0 a 3 d x 3 1 a 2 0 a 2 d x 2 1 a 1 0 a 1 d x 1 f ( x 1 𝐚 1 a 1 + x 2 𝐚 2 a 2 + x 3 𝐚 3 a 3 ) e - i 𝐊 𝐫 . h(\mathbf{K})=\frac{1}{a_{3}}\int_{0}^{a_{3}}dx_{3}\frac{1}{a_{2}}\int_{0}^{a_% {2}}dx_{2}\frac{1}{a_{1}}\int_{0}^{a_{1}}dx_{1}f\left(x_{1}\frac{\mathbf{a}_{1% }}{a_{1}}+x_{2}\frac{\mathbf{a}_{2}}{a_{2}}+x_{3}\frac{\mathbf{a}_{3}}{a_{3}}% \right)\cdot e^{-i\mathbf{K}\cdot\mathbf{r}}.
  48. 𝐫 = ( x , y , z ) = x 1 𝐚 1 a 1 + x 2 𝐚 2 a 2 + x 3 𝐚 3 a 3 , \mathbf{r}=(x,y,z)=x_{1}\frac{\mathbf{a}_{1}}{a_{1}}+x_{2}\frac{\mathbf{a}_{2}% }{a_{2}}+x_{3}\frac{\mathbf{a}_{3}}{a_{3}},
  49. [ x 1 x x 1 y x 1 z x 2 x x 2 y x 2 z x 3 x x 3 y x 3 z ] \begin{bmatrix}\dfrac{\partial x_{1}}{\partial x}&\dfrac{\partial x_{1}}{% \partial y}&\dfrac{\partial x_{1}}{\partial z}\\ \dfrac{\partial x_{2}}{\partial x}&\dfrac{\partial x_{2}}{\partial y}&\dfrac{% \partial x_{2}}{\partial z}\\ \dfrac{\partial x_{3}}{\partial x}&\dfrac{\partial x_{3}}{\partial y}&\dfrac{% \partial x_{3}}{\partial z}\end{bmatrix}
  50. a 1 a 2 a 3 𝐚 𝟏 ( 𝐚 𝟐 × 𝐚 𝟑 ) \frac{a_{1}a_{2}a_{3}}{\mathbf{a_{1}}\cdot(\mathbf{a_{2}}\times\mathbf{a_{3}})}
  51. d x 1 d x 2 d x 3 = a 1 a 2 a 3 𝐚 𝟏 ( 𝐚 𝟐 × 𝐚 𝟑 ) d x d y d z . dx_{1}\,dx_{2}\,dx_{3}=\frac{a_{1}a_{2}a_{3}}{\mathbf{a_{1}}\cdot(\mathbf{a_{2% }}\times\mathbf{a_{3}})}\cdot dx\,dy\,dz.
  52. h ( 𝐊 ) = 1 𝐚 𝟏 ( 𝐚 𝟐 × 𝐚 𝟑 ) C d 𝐫 f ( 𝐫 ) e - i 𝐊 𝐫 h(\mathbf{K})=\frac{1}{\mathbf{a_{1}}\cdot(\mathbf{a_{2}}\times\mathbf{a_{3}})% }\int_{C}d\mathbf{r}f(\mathbf{r})\cdot e^{-i\mathbf{K}\cdot\mathbf{r}}
  53. 𝐚 𝟏 ( 𝐚 𝟐 × 𝐚 𝟑 ) \mathbf{a_{1}}\cdot(\mathbf{a_{2}}\times\mathbf{a_{3}})
  54. e n = e i n x e_{n}=e^{inx}
  55. f , g = def 1 2 π - π π f ( x ) g ( x ) ¯ d x . \langle f,\,g\rangle\;\stackrel{\mathrm{def}}{=}\;\frac{1}{2\pi}\int_{-\pi}^{% \pi}f(x)\overline{g(x)}\,dx.
  56. f = n = - f , e n e n . f=\sum_{n=-\infty}^{\infty}\langle f,e_{n}\rangle\,e_{n}.
  57. - π π cos ( m x ) cos ( n x ) d x = π δ m n , m , n 1 , \int_{-\pi}^{\pi}\cos(mx)\,\cos(nx)\,dx=\pi\delta_{mn},\quad m,n\geq 1,\,
  58. - π π sin ( m x ) sin ( n x ) d x = π δ m n , m , n 1 \int_{-\pi}^{\pi}\sin(mx)\,\sin(nx)\,dx=\pi\delta_{mn},\quad m,n\geq 1
  59. - π π cos ( m x ) sin ( n x ) d x = 0 ; \int_{-\pi}^{\pi}\cos(mx)\,\sin(nx)\,dx=0;\,
  60. 2 \sqrt{2}
  61. 2 \sqrt{2}
  62. C k ( 𝕋 ) C^{k}(\mathbb{T})
  63. lim | n | f ^ ( n ) = 0 \lim_{|n|\rightarrow\infty}\hat{f}(n)=0
  64. lim n + a n = 0 \lim_{n\rightarrow+\infty}a_{n}=0
  65. lim n + b n = 0. \lim_{n\rightarrow+\infty}b_{n}=0.
  66. 2 ( 𝐙 ) \ell^{2}(\mathbf{Z})
  67. f C 1 ( 𝕋 ) f\in C^{1}(\mathbb{T})
  68. f ^ ( n ) \widehat{f^{\prime}}(n)
  69. f ^ ( n ) \hat{f}(n)
  70. f ^ ( n ) = i n f ^ ( n ) \widehat{f^{\prime}}(n)=in\hat{f}(n)
  71. f C k ( 𝕋 ) f\in C^{k}(\mathbb{T})
  72. f ( k ) ^ ( n ) = ( i n ) k f ^ ( n ) \widehat{f^{(k)}}(n)=(in)^{k}\hat{f}(n)
  73. f ( k ) ^ ( n ) \widehat{f^{(k)}}(n)
  74. | n | k f ^ ( n ) |n|^{k}\hat{f}(n)
  75. n = - | f ^ ( n ) | 2 = 1 2 π - π π | f ( x ) | 2 d x \sum_{n=-\infty}^{\infty}|\hat{f}(n)|^{2}=\frac{1}{2\pi}\int_{-\pi}^{\pi}|f(x)% |^{2}\,dx
  76. c 0 , c ± 1 , c ± 2 , c_{0},\,c_{\pm 1},\,c_{\pm 2},\ldots
  77. n = - | c n | 2 < \sum_{n=-\infty}^{\infty}|c_{n}|^{2}<\infty
  78. f L 2 ( [ - π , π ] ) f\in L^{2}([-\pi,\pi])
  79. f ^ ( n ) = c n \hat{f}(n)=c_{n}
  80. [ f * 2 π g ^ ] ( n ) = 2 π f ^ ( n ) g ^ ( n ) , [\widehat{f*_{2\pi}g}](n)=2\pi\cdot\hat{f}(n)\cdot\hat{g}(n),
  81. [ f * 2 π g ] ( x ) \displaystyle\left[f*_{2\pi}g\right](x)
  82. f ^ \hat{f}
  83. g ^ \hat{g}
  84. [ f g ^ ] ( n ) = [ f ^ * g ^ ] ( n ) . [\widehat{f\cdot g}](n)=[\hat{f}*\hat{g}](n).
  85. - \sum_{-\infty}^{\infty}
  86. f N ( x ) = n = - N N f ^ ( n ) e i n x . f_{N}(x)=\sum_{n=-N}^{N}\hat{f}(n)e^{inx}.
  87. p ( x ) = n = - N N p n e i n x . p(x)=\sum_{n=-N}^{N}p_{n}e^{inx}.
  88. f N - f 2 < p - f 2 , \|f_{N}-f\|_{2}<\|p-f\|_{2},
  89. g 2 = 1 2 π - π π | g ( x ) | 2 d x . \|g\|_{2}=\sqrt{{1\over 2\pi}\int_{-\pi}^{\pi}|g(x)|^{2}\,dx}.
  90. f N - f 2 \|f_{N}-f\|_{2}
  91. ( i n ) f ^ ( n ) (i\cdot n)\hat{f}(n)
  92. f C 1 ( 𝕋 ) f\in C^{1}(\mathbb{T})
  93. n 2 f ^ ( n ) n^{2}\hat{f}(n)
  94. sup x | f ( x ) - f N ( x ) | | n | > N | f ^ ( n ) | \sup_{x}|f(x)-f_{N}(x)|\leq\sum_{|n|>N}|\hat{f}(n)|
  95. { e i 2 π n x P } \mathcal{F}\left\{e^{i\frac{2\pi nx}{P}}\right\}

Fourier_transform.html

  1. m o d N ℤmodN
  2. f ^ \hat{f}
  3. f : f:\mathbb{R}\rightarrow\mathbb{C}
  4. f ^ ( ξ ) = - f ( x ) e - 2 π i x ξ d x , \hat{f}(\xi)=\int_{-\infty}^{\infty}f(x)\ e^{-2\pi ix\xi}\,dx,
  5. f f
  6. f ^ \hat{f}
  7. f ( x ) = - f ^ ( ξ ) e 2 π i ξ x d ξ , f(x)=\int_{-\infty}^{\infty}\hat{f}(\xi)\ e^{2\pi i\xi x}\,d\xi,
  8. f f
  9. f ^ \hat{f}
  10. f f
  11. f ^ \hat{f}
  12. c n = 1 T - T / 2 T / 2 f ( x ) e - 2 π i ( n / T ) x d x . c_{n}=\frac{1}{T}\int_{-T/2}^{T/2}f(x)\ e^{-2\pi i(n/T)x}dx.
  13. c n = ( 1 / T ) f ^ ( n / T ) c_{n}=(1/T)\hat{f}(n/T)
  14. f ( x ) = n = - c n e 2 π i ( n / T ) x = n = - f ^ ( ξ n ) e 2 π i ξ n x Δ ξ , f(x)=\sum_{n=-\infty}^{\infty}c_{n}\ e^{2\pi i(n/T)x}=\sum_{n=-\infty}^{\infty% }\hat{f}(\xi_{n})\ e^{2\pi i\xi_{n}x}\Delta\xi,
  15. f ^ ( 3 ) \hat{f}(3)
  16. f ^ ( 5 ) \hat{f}(5)
  17. - | f ( x ) | d x < . \int_{-\infty}^{\infty}|f(x)|\,dx<\infty.
  18. f ^ ( ξ ) \hat{f}(\xi)
  19. g ^ ( ξ ) \hat{g}(\xi)
  20. h ^ ( ξ ) \hat{h}(\xi)
  21. h ^ ( ξ ) = a f ^ ( ξ ) + b g ^ ( ξ ) . \hat{h}(\xi)=a\cdot\hat{f}(\xi)+b\cdot\hat{g}(\xi).
  22. h ( x ) = f ( x - x 0 ) , h(x)=f(x-x_{0}),
  23. h ^ ( ξ ) = e - i 2 π x 0 ξ f ^ ( ξ ) . \hat{h}(\xi)=e^{-i\,2\pi\,x_{0}\,\xi}\hat{f}(\xi).
  24. h ( x ) = e i 2 π x ξ 0 f ( x ) , h(x)=e^{i\,2\pi\,x\,\xi_{0}}f(x),
  25. h ^ ( ξ ) = f ^ ( ξ - ξ 0 ) . \hat{h}(\xi)=\hat{f}(\xi-\xi_{0}).
  26. h ^ ( ξ ) = 1 | a | f ^ ( ξ a ) . \hat{h}(\xi)=\frac{1}{|a|}\hat{f}\left(\frac{\xi}{a}\right).
  27. h ^ ( ξ ) = f ^ ( - ξ ) . \hat{h}(\xi)=\hat{f}(-\xi).
  28. h ( x ) = f ( x ) ¯ , h(x)=\overline{f(x)},
  29. h ^ ( ξ ) = f ^ ( - ξ ) ¯ . \hat{h}(\xi)=\overline{\hat{f}(-\xi)}.
  30. f ^ ( - ξ ) = f ^ ( ξ ) ¯ \hat{f}(-\xi)=\overline{\hat{f}(\xi)}
  31. f ^ \hat{f}
  32. f ^ ( - ξ ) = - f ^ ( ξ ) ¯ . \hat{f}(-\xi)=-\overline{\hat{f}(\xi)}.
  33. ξ = 0 \xi=0
  34. f ^ ( 0 ) = - f ( x ) d x . \hat{f}(0)=\int_{-\infty}^{\infty}f(x)\,dx.
  35. ξ = 0 \xi=0
  36. f ^ . \hat{f}.
  37. , \mathcal{F},
  38. ( f ) := f ^ , \mathcal{F}(f):=\hat{f},
  39. 2 ( f ) ( x ) = f ( - x ) , \mathcal{F}^{2}(f)(x)=f(-x),
  40. 4 ( f ) = f , \mathcal{F}^{4}(f)=f,
  41. 3 ( f ^ ) = f . \mathcal{F}^{3}(\hat{f})=f.
  42. 𝒫 \mathcal{P}
  43. 𝒫 [ f ] : t f ( - t ) , \mathcal{P}[f]\colon t\mapsto f(-t),
  44. 0 = Id , 1 = , 2 = 𝒫 , 4 = Id \mathcal{F}^{0}=\mathrm{Id},\qquad\mathcal{F}^{1}=\mathcal{F},\qquad\mathcal{F% }^{2}=\mathcal{P},\qquad\mathcal{F}^{4}=\mathrm{Id}
  45. 3 = - 1 = 𝒫 = 𝒫 \mathcal{F}^{3}=\mathcal{F}^{-1}=\mathcal{P}\circ\mathcal{F}=\mathcal{F}\circ% \mathcal{P}
  46. t t
  47. ξ \xi
  48. ξ \xi
  49. t t
  50. t t
  51. ξ \xi
  52. t t
  53. t t
  54. 2 π 2\pi
  55. ξ \xi
  56. t t
  57. ξ \xi
  58. t t
  59. t t
  60. ξ \xi
  61. ξ \xi
  62. t t
  63. t t
  64. ξ \xi
  65. ω = 2 π ξ \omega=2\pi\xi
  66. x ^ 1 \hat{x}_{1}
  67. x ^ \hat{x}
  68. x ^ 1 ( ω ) = x ^ ( ω 2 π ) = - x ( t ) e - i ω t d t \hat{x}_{1}(\omega)=\hat{x}\left({\omega\over 2\pi}\right)=\int_{-\infty}^{% \infty}x(t)e^{-i\omega t}dt
  69. x ( t ) = 1 2 π - x ^ 1 ( ω ) e i t ω d ω . x(t)={1\over{2\pi}}\int_{-\infty}^{\infty}\hat{x}_{1}(\omega)e^{it\omega}d\omega.
  70. 2 π \sqrt{2\pi}
  71. x ^ 2 ( ω ) = 1 2 π - x ( t ) e - i ω t d t , \hat{x}_{2}(\omega)={1\over\sqrt{2\pi}}\int_{-\infty}^{\infty}x(t)e^{-i\omega t% }dt,
  72. x ( t ) = 1 2 π - x ^ 2 ( ω ) e i t ω d ω . x(t)={1\over\sqrt{2\pi}}\int_{-\infty}^{\infty}\hat{x}_{2}(\omega)e^{it\omega}% d\omega.
  73. i i
  74. i i
  75. i i
  76. - i -i
  77. ϕ \phi
  78. f f
  79. X X
  80. x x
  81. 2 π 2\pi
  82. ϕ ( λ ) = - f ( x ) e i λ x d x \phi(\lambda)=\int_{-\infty}^{\infty}f(x)e^{i\lambda x}dx
  83. f ^ \hat{f}
  84. f ^ f 1 \|\hat{f}\|_{\infty}\leq\|f\|_{1}
  85. f ^ ( ξ ) 0 as | ξ | . \hat{f}(\xi)\to 0\,\text{ as }|\xi|\to\infty.
  86. f ^ \hat{f}
  87. f ^ \hat{f}
  88. f ( x ) = - f ^ ( ξ ) e 2 i π x ξ d ξ f(x)=\int_{-\infty}^{\infty}\hat{f}(\xi)e^{2i\pi x\xi}\,d\xi
  89. f ^ ( ξ ) \hat{f}(\xi)
  90. g ^ ( ξ ) \hat{g}(\xi)
  91. - f ( x ) g ( x ) ¯ d x = - f ^ ( ξ ) g ^ ( ξ ) ¯ d ξ , \int_{-\infty}^{\infty}f(x)\overline{g(x)}\,{\rm d}x=\int_{-\infty}^{\infty}% \hat{f}(\xi)\overline{\hat{g}(\xi)}\,d\xi,
  92. - | f ( x ) | 2 d x = - | f ^ ( ξ ) | 2 d ξ . \int_{-\infty}^{\infty}\left|f(x)\right|^{2}\,dx=\int_{-\infty}^{\infty}\left|% \hat{f}(\xi)\right|^{2}\,d\xi.
  93. f f
  94. n f ^ ( n ) = n f ( n ) . \sum_{n}\hat{f}(n)=\sum_{n}f(n).
  95. f ^ ( ξ ) = 2 π i ξ f ^ ( ξ ) . \widehat{f^{\prime}\;}(\xi)=2\pi i\xi\hat{f}(\xi).
  96. f ( n ) ^ ( ξ ) = ( 2 π i ξ ) n f ^ ( ξ ) . \widehat{f^{(n)}}(\xi)=(2\pi i\xi)^{n}\hat{f}(\xi).
  97. f ( x ) f(x)
  98. f ^ ( ξ ) \hat{f}(\xi)
  99. | ξ | |\xi|\to\infty
  100. f ( x ) f(x)
  101. | x | |x|\to\infty
  102. f ^ ( ξ ) \hat{f}(\xi)
  103. f ^ ( ξ ) \hat{f}(\xi)
  104. g ^ ( ξ ) \hat{g}(\xi)
  105. f ^ ( ξ ) \hat{f}(\xi)
  106. g ^ ( ξ ) \hat{g}(\xi)
  107. h ( x ) = ( f * g ) ( x ) = - f ( y ) g ( x - y ) d y , h(x)=(f*g)(x)=\int_{-\infty}^{\infty}f(y)g(x-y)\,dy,
  108. h ^ ( ξ ) = f ^ ( ξ ) g ^ ( ξ ) . \hat{h}(\xi)=\hat{f}(\xi)\cdot\hat{g}(\xi).
  109. g ^ ( ξ ) \hat{g}(\xi)
  110. p ^ ( ξ ) \hat{p}(\xi)
  111. q ^ ( ξ ) \hat{q}(\xi)
  112. h ( x ) = ( f g ) ( x ) = - f ( y ) ¯ g ( x + y ) d y h(x)=(f\star g)(x)=\int_{-\infty}^{\infty}\overline{f(y)}\,g(x+y)\,dy
  113. h ^ ( ξ ) = f ^ ( ξ ) ¯ g ^ ( ξ ) . \hat{h}(\xi)=\overline{\hat{f}(\xi)}\,\cdot\,\hat{g}(\xi).
  114. h ( x ) = ( f f ) ( x ) = - f ( y ) ¯ f ( x + y ) d y h(x)=(f\star f)(x)=\int_{-\infty}^{\infty}\overline{f(y)}f(x+y)\,dy
  115. h ^ ( ξ ) = f ^ ( ξ ) ¯ f ^ ( ξ ) = | f ^ ( ξ ) | 2 . \hat{h}(\xi)=\overline{\hat{f}(\xi)}\,\hat{f}(\xi)=|\hat{f}(\xi)|^{2}.
  116. ψ n ( x ) = 2 1 / 4 n ! e - π x 2 He n ( 2 x π ) , {\psi}_{n}(x)=\frac{2^{1/4}}{\sqrt{n!}}\,e^{-\pi x^{2}}\mathrm{He}_{n}(2x\sqrt% {\pi}),
  117. He n ( x ) = ( - 1 ) n e x 2 2 ( d d x ) n e - x 2 2 \mathrm{He}_{n}(x)=(-1)^{n}e^{\frac{x^{2}}{2}}\left(\frac{d}{dx}\right)^{n}e^{% -\frac{x^{2}}{2}}
  118. ψ ^ n ( ξ ) = ( - i ) n ψ n ( ξ ) \hat{\psi}_{n}(\xi)=(-i)^{n}{\psi}_{n}(\xi)
  119. f ^ ( ξ ) = - e - 2 π i ξ t f ( t ) d t \hat{f}(\xi)=\int_{-\infty}^{\infty}e^{-2\pi i\xi t}f(t)dt
  120. ξ \xi
  121. f f
  122. ξ = σ + i τ \xi=\sigma+i\tau
  123. f f
  124. n n
  125. n n
  126. f ^ ( σ + i τ ) \hat{f}(\sigma+i\tau)
  127. a > 0 a>0
  128. n 0 n\geq 0
  129. | ξ n f ^ ( ξ ) | C e a | τ | |\xi^{n}\hat{f}(\xi)|\leq Ce^{a|\tau|}
  130. C C
  131. f f
  132. [ - a , a ] [-a,a]
  133. f ^ \hat{f}
  134. σ \sigma
  135. τ \tau
  136. τ \tau
  137. σ \sigma
  138. f f
  139. L 2 L^{2}
  140. n = 0 n=0
  141. f f
  142. t 0 t\geq 0
  143. f f
  144. f ^ \hat{f}
  145. τ < 0 \tau<0
  146. τ \tau
  147. f ^ ( ξ ) \hat{f}(\xi)
  148. F ( s ) F(s)
  149. s s
  150. f f
  151. f ( t ) f(t)
  152. | f ( t ) | < C e a | t | |f(t)|<Ce^{a|t|}
  153. C , a 0 C,a\geq 0
  154. f ^ ( i τ ) = - e 2 π τ t f ( t ) d t , \hat{f}(i\tau)=\int_{-\infty}^{\infty}e^{2\pi\tau t}f(t)dt,
  155. 2 π τ < - a 2\pi\tau<-a
  156. f f
  157. F ( s ) = 0 f ( t ) e - s t d t . F(s)=\int_{0}^{\infty}f(t)e^{-st}dt.
  158. f f
  159. f ^ ( i τ ) = F ( - 2 π τ ) . \hat{f}(i\tau)=F(-2\pi\tau).
  160. s = 2 π i ξ s=2\pi i\xi
  161. f ^ \hat{f}
  162. a τ b a\leq\tau\leq b
  163. - f ^ ( σ + i a ) e 2 π i ξ t d σ = - f ^ ( σ + i b ) e 2 π i ξ t d σ \int_{-\infty}^{\infty}\hat{f}(\sigma+ia)e^{2\pi i\xi t}d\sigma=\int_{-\infty}% ^{\infty}\hat{f}(\sigma+ib)e^{2\pi i\xi t}d\sigma
  164. f ( t ) = 0 f(t)=0
  165. t < 0 t<0
  166. | f ( t ) | < C e a | t | |f(t)|<Ce^{a|t|}
  167. C , a > 0 C,a>0
  168. f ( t ) = - f ^ ( σ + i τ ) e 2 π i ξ t d σ , f(t)=\int_{-\infty}^{\infty}\hat{f}(\sigma+i\tau)e^{2\pi i\xi t}d\sigma,
  169. τ < - a 2 π \tau<-{a\over 2\pi}
  170. f ( t ) = 1 2 π i b - i b + i F ( s ) e s t d s f(t)=\frac{1}{2\pi i}\int_{b-i\infty}^{b+i\infty}F(s)e^{st}ds
  171. b > a b>a
  172. F ( s ) F(s)
  173. f ( t ) f(t)
  174. f ( t ) e - a t f(t)e^{-at}
  175. L 1 L^{1}
  176. t t
  177. f f
  178. L 2 L^{2}
  179. f ^ ( s y m b o l ξ ) = ( f ) ( s y m b o l ξ ) = \R n f ( 𝐱 ) e - 2 π i 𝐱 \cdotsymbol ξ d 𝐱 \hat{f}(symbol{\xi})=\mathcal{F}(f)(symbol{\xi})=\int_{\R^{n}}f(\mathbf{x})e^{% -2\pi i\mathbf{x}\cdotsymbol{\xi}}\,d\mathbf{x}
  180. 𝐱 , s y m b o l ξ \left\langle\mathbf{x},symbol\xi\right\rangle
  181. f ^ ( ξ ) \hat{f}(\xi)
  182. - | f ( x ) | 2 d x = 1. \int_{-\infty}^{\infty}|f(x)|^{2}\,dx=1.
  183. f ^ ( ξ ) \hat{f}(\xi)
  184. D 0 ( f ) = - x 2 | f ( x ) | 2 d x . D_{0}(f)=\int_{-\infty}^{\infty}x^{2}|f(x)|^{2}\,dx.
  185. D 0 ( f ) D 0 ( f ^ ) 1 16 π 2 D_{0}(f)D_{0}(\hat{f})\geq\frac{1}{16\pi^{2}}
  186. f ( x ) = C 1 e - π x 2 / σ 2 f(x)=C_{1}\,e^{{-\pi x^{2}}/{\sigma^{2}}}
  187. f ^ ( ξ ) = σ C 1 e - π σ 2 ξ 2 \hat{f}(\xi)=\sigma C_{1}\,e^{-\pi\sigma^{2}\xi^{2}}
  188. C 1 = 2 4 / σ C_{1}=\sqrt[4]{2}/\sqrt{\sigma}
  189. ( - ( x - x 0 ) 2 | f ( x ) | 2 d x ) ( - ( ξ - ξ 0 ) 2 | f ^ ( ξ ) | 2 d ξ ) 1 16 π 2 \left(\int_{-\infty}^{\infty}(x-x_{0})^{2}|f(x)|^{2}\,dx\right)\left(\int_{-% \infty}^{\infty}(\xi-\xi_{0})^{2}|\hat{f}(\xi)|^{2}\,d\xi\right)\geq\frac{1}{1% 6\pi^{2}}
  190. H ( | f | 2 ) + H ( | f ^ | 2 ) log ( e / 2 ) H(|f|^{2})+H(|\hat{f}|^{2})\geq\log(e/2)
  191. H ( p ) = - - p ( x ) log ( p ( x ) ) d x H(p)=-\int_{-\infty}^{\infty}p(x)\log(p(x))dx
  192. f f
  193. λ \lambda
  194. f ( t ) = 0 [ a ( λ ) cos 2 π λ t + b ( λ ) sin 2 π λ t ] d λ . f(t)=\int_{0}^{\infty}\left[a(\lambda)\cos 2\pi\lambda t+b(\lambda)\sin 2\pi% \lambda t\right]d\lambda.
  195. a a
  196. b b
  197. a ( λ ) = 2 - f ( t ) cos 2 π λ t d t a(\lambda)=2\int_{-\infty}^{\infty}f(t)\cos 2\pi\lambda tdt
  198. b ( λ ) = 2 - f ( t ) sin 2 π λ t d t . b(\lambda)=2\int_{-\infty}^{\infty}f(t)\sin 2\pi\lambda tdt.
  199. a a
  200. b b
  201. f ( t ) = 2 0 - f ( τ ) cos 2 π λ ( τ - t ) d τ d λ . f(t)=2\int_{0}^{\infty}\int_{-\infty}^{\infty}f(\tau)\cos 2\pi\lambda(\tau-t)d% \tau d\lambda.
  202. f ^ ( ξ ) = i - k f ( ξ ) \hat{f}(\xi)=i^{-k}f(\xi)
  203. f ^ ( ξ ) = F 0 ( | ξ | ) P ( ξ ) \hat{f}(\xi)=F_{0}(|\xi|)P(\xi)
  204. F 0 ( r ) = 2 π i - k r - ( n + 2 k - 2 ) / 2 0 f 0 ( s ) J ( n + 2 k - 2 ) / 2 ( 2 π r s ) s ( n + 2 k ) / 2 d s . F_{0}(r)=2\pi i^{-k}r^{-(n+2k-2)/2}\int_{0}^{\infty}f_{0}(s)J_{(n+2k-2)/2}(2% \pi rs)s^{(n+2k)/2}\,ds.
  205. f R ( x ) = E R f ^ ( ξ ) e 2 π i x ξ d ξ , x 𝐑 n . f_{R}(x)=\int_{E_{R}}\hat{f}(\xi)e^{2\pi ix\cdot\xi}\,d\xi,\quad x\in\mathbf{R% }^{n}.
  206. f ^ ( ξ ) = 𝐑 n f ( x ) e - 2 π i ξ x d x \hat{f}(\xi)=\int_{\mathbf{R}^{n}}f(x)e^{-2\pi i\xi\cdot x}\,dx
  207. \mathcal{F}
  208. | f ^ ( ξ ) | 𝐑 n | f ( x ) | d x , |\hat{f}(\xi)|\leq\int_{\mathbf{R}^{n}}|f(x)|\,dx,
  209. f ^ ( ξ ) = lim R | x | R f ( x ) e - 2 π i x ξ d x \hat{f}(\xi)=\lim_{R\to\infty}\int_{|x|\leq R}f(x)e^{-2\pi ix\cdot\xi}\,dx
  210. \mathcal{F}
  211. 𝐑 n f ( x ) g ( x ) d x = 𝐑 n f ( x ) g ( x ) d x . \int_{\mathbf{R}^{n}}f(x)\mathcal{F}g(x)\,dx=\int_{\mathbf{R}^{n}}\mathcal{F}f% (x)g(x)\,dx.
  212. q = p / ( p - 1 ) q=p/(p-1)
  213. f ^ \hat{f}
  214. g ^ \hat{g}
  215. 𝐑 n f ^ ( x ) g ( x ) d x = 𝐑 n f ( x ) g ^ ( x ) d x . \int_{\mathbf{R}^{n}}\hat{f}(x)g(x)\,dx=\int_{\mathbf{R}^{n}}f(x)\hat{g}(x)\,dx.
  216. T f ( φ ) = 𝐑 n f ( x ) φ ( x ) d x T_{f}(\varphi)=\int_{\mathbf{R}^{n}}f(x)\varphi(x)\,dx
  217. T ^ f \hat{T}_{f}
  218. T ^ f ( φ ) = T f ( φ ^ ) \hat{T}_{f}(\varphi)=T_{f}(\hat{\varphi})
  219. μ ^ ( ξ ) = 𝐑 n e - 2 π i x ξ d μ . \hat{\mu}(\xi)=\int_{\mathbf{R}^{n}}\mathrm{e}^{-2\pi ix\cdot\xi}\,d\mu.
  220. G ^ \hat{G}
  221. f ^ ( ξ ) = G ξ ( x ) f ( x ) d μ for any ξ G ^ . \hat{f}(\xi)=\int_{G}\xi(x)f(x)\,d\mu\qquad\,\text{for any }\xi\in\hat{G}.
  222. f ^ ( ξ ) \hat{f}(\xi)
  223. G ^ \hat{G}
  224. f * ( g ) = f ( g - 1 ) ¯ . f^{*}(g)=\overline{f(g^{-1})}.
  225. a ( φ φ ( a ) ) a\mapsto(\varphi\mapsto\varphi(a))
  226. μ ^ ξ , η H σ = G U ¯ g ( σ ) ξ , η d μ ( g ) \langle\hat{\mu}\xi,\eta\rangle_{H_{\sigma}}=\int_{G}\langle\overline{U}^{(% \sigma)}_{g}\xi,\eta\rangle\,d\mu(g)
  227. U ¯ ( σ ) \overline{U}^{(\sigma)}
  228. d μ = f d λ d\mu=fd\lambda
  229. μ μ ^ \mu\mapsto\hat{\mu}
  230. E = sup σ Σ E σ \|E\|=\sup_{\sigma\in\Sigma}\|E_{\sigma}\|
  231. f * ( g ) = f ( g - 1 ) ¯ , f^{*}(g)=\overline{f(g^{-1})},
  232. f ( g ) = σ Σ d σ tr ( f ^ ( σ ) U g ( σ ) ) f(g)=\sum_{\sigma\in\Sigma}d_{\sigma}\operatorname{tr}(\hat{f}(\sigma)U^{(% \sigma)}_{g})
  233. 2 y ( x , t ) 2 x = y ( x , t ) t . {\partial^{2}y(x,t)\over\partial^{2}x}={\partial y(x,t)\over\partial t}.
  234. 2 y ( x , t ) 2 x = 2 y ( x , t ) 2 t . {\partial^{2}y(x,t)\over\partial^{2}x}={\partial^{2}y(x,t)\over\partial^{2}t}.
  235. y ( x , 0 ) = f ( x ) , y ( x , t ) t = g ( x ) . y(x,0)=f(x),{\partial y(x,t)\over\partial t}=g(x).
  236. f f
  237. g g
  238. y y
  239. y ^ \hat{y}
  240. y ^ \hat{y}
  241. y y
  242. cos 2 π ξ ( x ± t ) or sin 2 π ξ ( x ± t ) \cos 2\pi\xi(x\pm t)\mbox{ or }~{}\sin 2\pi\xi(x\pm t)
  243. y ( x , t ) = 0 a + ( ξ ) cos 2 π ξ ( x + t ) + a - ( ξ ) cos 2 π ξ ( x - t ) + b + ( ξ ) sin 2 π ξ ( x + t ) + b - ( ξ ) sin 2 π ξ ( x - t ) d ξ y(x,t)=\int_{0}^{\infty}a_{+}(\xi)\cos 2\pi\xi(x+t)+a_{-}(\xi)\cos 2\pi\xi(x-t% )+b_{+}(\xi)\sin 2\pi\xi(x+t)+b_{-}(\xi)\sin 2\pi\xi(x-t)d\xi
  244. a + a_{+}
  245. a - a_{-}
  246. b + b_{+}
  247. b - b_{-}
  248. a ± a_{\pm}
  249. b ± b_{\pm}
  250. x x
  251. a ± a_{\pm}
  252. b ± b_{\pm}
  253. y y
  254. t = 0 t=0
  255. t = 0 t=0
  256. x x
  257. 2 - y ( x , 0 ) cos 2 π ξ x d x = a + + a - 2\int_{-\infty}^{\infty}y(x,0)\cos 2\pi\xi xdx=a_{+}+a_{-}
  258. 2 - y ( x , 0 ) sin 2 π ξ x d x = b + + b - . 2\int_{-\infty}^{\infty}y(x,0)\sin 2\pi\xi xdx=b_{+}+b_{-}.
  259. y y
  260. t t
  261. 2 - y ( u , 0 ) t sin ( 2 π ξ x ) d x = ( 2 π ξ ) ( - a + + a - ) 2\int_{-\infty}^{\infty}{\partial y(u,0)\over\partial t}\sin(2\pi\xi x)dx=(2% \pi\xi)(-a_{+}+a_{-})
  262. 2 - y ( u , 0 ) t cos ( 2 π ξ x ) d x = ( 2 π ξ ) ( b + - b - ) . 2\int_{-\infty}^{\infty}{\partial y(u,0)\over\partial t}\cos(2\pi\xi x)dx=(2% \pi\xi)(b_{+}-b_{-}).
  263. a ± a_{\pm}
  264. b ± b_{\pm}
  265. ξ \xi
  266. ξ \xi
  267. f f
  268. g g
  269. a ± a_{\pm}
  270. b ± b_{\pm}
  271. f f
  272. g g
  273. x x
  274. t t
  275. y ^ \hat{y}
  276. y ( x , t ) y(x,t)
  277. L 1 L^{1}
  278. x x
  279. 2 π i ξ 2\pi i\xi
  280. t t
  281. 2 π i f 2\pi if
  282. f f
  283. y ^ \hat{y}
  284. ξ 2 y ^ ( ξ , f ) = f 2 y ^ ( ξ , f ) . \xi^{2}\hat{y}(\xi,f)=f^{2}\hat{y}(\xi,f).
  285. y ^ ( ξ , f ) = 0 \hat{y}(\xi,f)=0
  286. ξ = ± f \xi=\pm f
  287. f ^ = δ ( ξ ± f ) \hat{f}=\delta(\xi\pm f)
  288. ξ 2 - f 2 = 0 \xi^{2}-f^{2}=0
  289. ξ = f \xi=f
  290. ξ = - f \xi=-f
  291. ϕ \phi
  292. y ^ ϕ ( ξ , f ) d ξ d f = s + ϕ ( ξ , ξ ) d ξ + s - ϕ ( ξ , - ξ ) d ξ , \int\int\hat{y}\phi(\xi,f)d\xi df=\int s_{+}\phi(\xi,\xi)d\xi+\int s_{-}\phi(% \xi,-\xi)d\xi,
  293. s + s_{+}
  294. s - s_{-}
  295. ϕ ( ξ , f ) = e 2 π i ( x ξ + t f ) \phi(\xi,f)=e^{2\pi i(x\xi+tf)}
  296. y ( x , 0 ) = { s + ( ξ ) + s - ( ξ ) } e 2 π i ξ x + 0 d ξ y(x,0)=\int\{s_{+}(\xi)+s_{-}(\xi)\}e^{2\pi i\xi x+0}d\xi
  297. y ( x , 0 ) t = { s + ( ξ ) - s - ( ξ ) } 2 π i ξ e 2 π i ξ x + 0 d ξ . {\partial y(x,0)\over\partial t}=\int\{s_{+}(\xi)-s_{-}(\xi)\}2\pi i\xi e^{2% \pi i\xi x+0}d\xi.
  298. x x
  299. x x
  300. s ± s_{\pm}
  301. L 1 L^{1}
  302. L 2 L^{2}
  303. q q
  304. p p
  305. q q
  306. p p
  307. p p
  308. q q
  309. p p
  310. q q
  311. p p
  312. q q
  313. q q
  314. p p
  315. ϕ ( p ) = ψ ( q ) e 2 π i p q h d q . \phi(p)=\int\psi(q)e^{2\pi ipq\over h}dq.
  316. L 2 L^{2}
  317. L 2 L^{2}
  318. q q
  319. p p
  320. 2 x 2 ψ ( x , t ) = i h 2 π t ψ ( x , t ) . {\partial^{2}\over\partial x^{2}}\psi(x,t)=i\frac{h}{2\pi}{\partial\over% \partial t}\psi(x,t).
  321. i i
  322. V ( x ) V(x)
  323. 2 x 2 ψ ( x , t ) + V ( x ) ψ ( x , t ) = i h 2 π t ψ ( x , t ) . {\partial^{2}\over\partial x^{2}}\psi(x,t)+V(x)\psi(x,t)=i\frac{h}{2\pi}{% \partial\over\partial t}\psi(x,t).
  324. ψ \psi
  325. t = 0 t=0
  326. ( 2 x 2 + 1 ) ψ ( x , t ) = 2 t 2 ψ ( x , t ) . ({\partial^{2}\over\partial x^{2}}+1)\psi(x,t)={\partial^{2}\over\partial t^{2% }}\psi(x,t).
  327. R R
  328. f f
  329. \R f ( τ ) = lim T 1 2 T - T T f ( t ) f ( t + τ ) d t . \R_{f}(\tau)=\lim_{T\rightarrow\infty}{1\over 2T}\int_{-T}^{T}f(t)f(t+\tau)dt.
  330. τ \tau
  331. f f
  332. f f
  333. R R
  334. τ \tau
  335. τ = \tau=
  336. f f
  337. f f
  338. f ( t ) f(t)
  339. t t
  340. P f ( ξ ) = - R f ( τ ) e - 2 π i ξ τ d τ . P_{f}(\xi)=\int_{-\infty}^{\infty}R_{f}(\tau)e^{-2\pi i\xi\tau}d\tau.
  341. f f
  342. f f
  343. P P
  344. ξ \xi
  345. f ^ ( ξ ) \hat{f}(\xi)
  346. f ~ ( ξ ) , f ~ ( ω ) , F ( ξ ) , ( f ) ( ξ ) , ( f ) ( ξ ) , ( f ) , ( ω ) , F ( ω ) , ( j ω ) , { f } , ( f ( t ) ) , { f ( t ) } . \tilde{f}(\xi),\ \tilde{f}(\omega),\ F(\xi),\ \mathcal{F}\left(f\right)(\xi),% \ \left(\mathcal{F}f\right)(\xi),\ \mathcal{F}(f),\ \mathcal{F}(\omega),\ F(% \omega),\ \mathcal{F}(j\omega),\ \mathcal{F}\{f\},\ \mathcal{F}\left(f(t)% \right),\ \mathcal{F}\{f(t)\}.
  347. f ^ ( ξ ) \hat{f}(\xi)
  348. f ^ ( ξ ) = A ( ξ ) e i φ ( ξ ) \hat{f}(\xi)=A(\xi)e^{i\varphi(\xi)}
  349. A ( ξ ) = | f ^ ( ξ ) | , A(\xi)=|\hat{f}(\xi)|,
  350. φ ( ξ ) = arg ( f ^ ( ξ ) ) , \varphi(\xi)=\arg\big(\hat{f}(\xi)\big),
  351. f ( x ) = - A ( ξ ) e i ( 2 π ξ x + φ ( ξ ) ) d ξ , f(x)=\int_{-\infty}^{\infty}A(\xi)\ e^{i(2\pi\xi x+\varphi(\xi))}\,d\xi,
  352. \mathcal{F}
  353. ( f ) \mathcal{F}(f)
  354. \mathcal{F}
  355. f \mathcal{F}f
  356. ( f ) \mathcal{F}(f)
  357. f ( ξ ) \mathcal{F}f(\xi)
  358. ( f ) ( ξ ) (\mathcal{F}f)(\xi)
  359. \mathcal{F}
  360. ( f ( x ) ) \mathcal{F}(f(x))
  361. ( rect ( x ) ) = sinc ( ξ ) \mathcal{F}(\mathrm{rect}(x))=\mathrm{sinc}(\xi)
  362. ( f ( x + x 0 ) ) = ( f ( x ) ) e 2 π i ξ x 0 \mathcal{F}(f(x+x_{0}))=\mathcal{F}(f(x))e^{2\pi i\xi x_{0}}
  363. ω = 2 π ξ , \omega=2\pi\xi,
  364. f ^ ( ω ) = 𝐑 n f ( x ) e - i ω x d x . \hat{f}(\omega)=\int_{\mathbf{R}^{n}}f(x)e^{-i\omega\cdot x}\,dx.
  365. f ( x ) = 1 ( 2 π ) n 𝐑 n f ^ ( ω ) e i ω x d ω . f(x)=\frac{1}{(2\pi)^{n}}\int_{\mathbf{R}^{n}}\hat{f}(\omega)e^{i\omega\cdot x% }\,d\omega.
  366. f ^ ( ω ) = 1 ( 2 π ) n / 2 𝐑 n f ( x ) e - i ω x d x , \hat{f}(\omega)=\frac{1}{(2\pi)^{n/2}}\int_{\mathbf{R}^{n}}f(x)e^{-i\omega% \cdot x}\,dx,
  367. f ( x ) = 1 ( 2 π ) n / 2 𝐑 n f ^ ( ω ) e i ω x d ω . f(x)=\frac{1}{(2\pi)^{n/2}}\int_{\mathbf{R}^{n}}\hat{f}(\omega)e^{i\omega\cdot x% }\,d\omega.
  368. f ^ 1 ( ξ ) = def 𝐑 n f ( x ) e - 2 π i x ξ d x = f ^ 2 ( 2 π ξ ) = ( 2 π ) n / 2 f ^ 3 ( 2 π ξ ) \displaystyle\hat{f}_{1}(\xi)\ \stackrel{\mathrm{def}}{=}\ \int_{\mathbf{R}^{n% }}f(x)e^{-2\pi ix\cdot\xi}\,dx=\hat{f}_{2}(2\pi\xi)=(2\pi)^{n/2}\hat{f}_{3}(2% \pi\xi)
  369. f ( x ) = 𝐑 n f ^ 1 ( ξ ) e 2 π i x ξ d ξ \displaystyle f(x)=\int_{\mathbf{R}^{n}}\hat{f}_{1}(\xi)e^{2\pi ix\cdot\xi}\,d\xi
  370. f ^ 3 ( ω ) = def 1 ( 2 π ) n / 2 𝐑 n f ( x ) e - i ω x d x = 1 ( 2 π ) n / 2 f ^ 1 ( ω 2 π ) = 1 ( 2 π ) n / 2 f ^ 2 ( ω ) \displaystyle\hat{f}_{3}(\omega)\stackrel{\mathrm{def}}{{}={}}\frac{1}{(2\pi)^% {n/2}}\int_{\mathbf{R}^{n}}f(x)e^{-i\omega\cdot x}\,dx=\frac{1}{(2\pi)^{n/2}}% \hat{f}_{1}\!\left(\frac{\omega}{2\pi}\right)=\frac{1}{(2\pi)^{n/2}}\hat{f}_{2% }(\omega)
  371. f ( x ) = 1 ( 2 π ) n / 2 𝐑 n f ^ 3 ( ω ) e i ω x d ω \displaystyle f(x)=\frac{1}{(2\pi)^{n/2}}\int_{\mathbf{R}^{n}}\hat{f}_{3}(% \omega)e^{i\omega\cdot x}\,d\omega
  372. f ^ 2 ( ω ) = def 𝐑 n f ( x ) e - i ω x d x = f ^ 1 ( ω 2 π ) = ( 2 π ) n / 2 f ^ 3 ( ω ) \displaystyle\hat{f}_{2}(\omega)\ \stackrel{\mathrm{def}}{=}\int_{\mathbf{R}^{% n}}f(x)e^{-i\omega\cdot x}\,dx=\hat{f}_{1}\!\left(\frac{\omega}{2\pi}\right)=(% 2\pi)^{n/2}\hat{f}_{3}(\omega)
  373. f ( x ) = 1 ( 2 π ) n 𝐑 n f ^ 2 ( ω ) e i ω x d ω \displaystyle f(x)=\frac{1}{(2\pi)^{n}}\int_{\mathbf{R}^{n}}\hat{f}_{2}(\omega% )e^{i\omega\cdot x}\,d\omega
  374. E ( e i t X ) = e i t x d μ X ( x ) E(e^{it\cdot X})=\int e^{it\cdot x}d\mu_{X}(x)
  375. f ^ \hat{f}
  376. g ^ \hat{g}
  377. h ^ \hat{h}
  378. f ( x ) \displaystyle f(x)\,
  379. f ^ ( ξ ) = \displaystyle\hat{f}(\xi)=
  380. - f ( x ) e - 2 π i x ξ d x \displaystyle\int_{-\infty}^{\infty}f(x)e^{-2\pi ix\xi}\,dx
  381. f ^ ( ω ) = \displaystyle\hat{f}(\omega)=
  382. 1 2 π - f ( x ) e - i ω x d x \displaystyle\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(x)e^{-i\omega x}\,dx
  383. f ^ ( ν ) = \displaystyle\hat{f}(\nu)=
  384. - f ( x ) e - i ν x d x \displaystyle\int_{-\infty}^{\infty}f(x)e^{-i\nu x}\,dx
  385. a f ( x ) + b g ( x ) \displaystyle a\cdot f(x)+b\cdot g(x)\,
  386. a f ^ ( ξ ) + b g ^ ( ξ ) \displaystyle a\cdot\hat{f}(\xi)+b\cdot\hat{g}(\xi)\,
  387. a f ^ ( ω ) + b g ^ ( ω ) \displaystyle a\cdot\hat{f}(\omega)+b\cdot\hat{g}(\omega)\,
  388. a f ^ ( ν ) + b g ^ ( ν ) \displaystyle a\cdot\hat{f}(\nu)+b\cdot\hat{g}(\nu)\,
  389. f ( x - a ) \displaystyle f(x-a)\,
  390. e - 2 π i a ξ f ^ ( ξ ) \displaystyle e^{-2\pi ia\xi}\hat{f}(\xi)\,
  391. e - i a ω f ^ ( ω ) \displaystyle e^{-ia\omega}\hat{f}(\omega)\,
  392. e - i a ν f ^ ( ν ) \displaystyle e^{-ia\nu}\hat{f}(\nu)\,
  393. e 2 π i a x f ( x ) \displaystyle e^{2\pi iax}f(x)\,
  394. f ^ ( ξ - a ) \displaystyle\hat{f}\left(\xi-a\right)\,
  395. f ^ ( ω - 2 π a ) \displaystyle\hat{f}(\omega-2\pi a)\,
  396. f ^ ( ν - 2 π a ) \displaystyle\hat{f}(\nu-2\pi a)\,
  397. f ( a x ) \displaystyle f(ax)\,
  398. 1 | a | f ^ ( ξ a ) \displaystyle\frac{1}{|a|}\hat{f}\left(\frac{\xi}{a}\right)\,
  399. 1 | a | f ^ ( ω a ) \displaystyle\frac{1}{|a|}\hat{f}\left(\frac{\omega}{a}\right)\,
  400. 1 | a | f ^ ( ν a ) \displaystyle\frac{1}{|a|}\hat{f}\left(\frac{\nu}{a}\right)\,
  401. | a | \displaystyle|a|\,
  402. f ( a x ) \displaystyle f(ax)\,
  403. 1 | a | f ^ ( ω a ) \displaystyle\frac{1}{|a|}\hat{f}\left(\frac{\omega}{a}\right)\,
  404. f ^ ( x ) \displaystyle\hat{f}(x)\,
  405. f ( - ξ ) \displaystyle f(-\xi)\,
  406. f ( - ω ) \displaystyle f(-\omega)\,
  407. 2 π f ( - ν ) \displaystyle 2\pi f(-\nu)\,
  408. f ^ \hat{f}
  409. x x
  410. ξ \xi
  411. ω \omega
  412. ν \nu
  413. d n f ( x ) d x n \displaystyle\frac{d^{n}f(x)}{dx^{n}}\,
  414. ( 2 π i ξ ) n f ^ ( ξ ) \displaystyle(2\pi i\xi)^{n}\hat{f}(\xi)\,
  415. ( i ω ) n f ^ ( ω ) \displaystyle(i\omega)^{n}\hat{f}(\omega)\,
  416. ( i ν ) n f ^ ( ν ) \displaystyle(i\nu)^{n}\hat{f}(\nu)\,
  417. x n f ( x ) \displaystyle x^{n}f(x)\,
  418. ( i 2 π ) n d n f ^ ( ξ ) d ξ n \displaystyle\left(\frac{i}{2\pi}\right)^{n}\frac{d^{n}\hat{f}(\xi)}{d\xi^{n}}\,
  419. i n d n f ^ ( ω ) d ω n \displaystyle i^{n}\frac{d^{n}\hat{f}(\omega)}{d\omega^{n}}
  420. i n d n f ^ ( ν ) d ν n \displaystyle i^{n}\frac{d^{n}\hat{f}(\nu)}{d\nu^{n}}
  421. ( f * g ) ( x ) \displaystyle(f*g)(x)\,
  422. f ^ ( ξ ) g ^ ( ξ ) \displaystyle\hat{f}(\xi)\hat{g}(\xi)\,
  423. 2 π f ^ ( ω ) g ^ ( ω ) \displaystyle\sqrt{2\pi}\hat{f}(\omega)\hat{g}(\omega)\,
  424. f ^ ( ν ) g ^ ( ν ) \displaystyle\hat{f}(\nu)\hat{g}(\nu)\,
  425. f * g \displaystyle f*g\,
  426. f f
  427. g g
  428. f ( x ) g ( x ) \displaystyle f(x)g(x)\,
  429. ( f ^ * g ^ ) ( ξ ) \displaystyle(\hat{f}*\hat{g})(\xi)\,
  430. ( f ^ * g ^ ) ( ω ) 2 π \displaystyle(\hat{f}*\hat{g})(\omega)\over\sqrt{2\pi}\,
  431. 1 2 π ( f ^ * g ^ ) ( ν ) \displaystyle\frac{1}{2\pi}(\hat{f}*\hat{g})(\nu)\,
  432. f ( x ) \displaystyle f(x)\,
  433. f ^ ( - ξ ) = f ^ ( ξ ) ¯ \displaystyle\hat{f}(-\xi)=\overline{\hat{f}(\xi)}\,
  434. f ^ ( - ω ) = f ^ ( ω ) ¯ \displaystyle\hat{f}(-\omega)=\overline{\hat{f}(\omega)}\,
  435. f ^ ( - ν ) = f ^ ( ν ) ¯ \displaystyle\hat{f}(-\nu)=\overline{\hat{f}(\nu)}\,
  436. z ¯ \displaystyle\overline{z}\,
  437. f ( x ) \displaystyle f(x)\,
  438. f ^ ( ω ) \displaystyle\hat{f}(\omega)
  439. f ^ ( ξ ) \displaystyle\hat{f}(\xi)
  440. f ^ ( ν ) \displaystyle\hat{f}(\nu)\,
  441. f ( x ) \displaystyle f(x)\,
  442. f ^ ( ω ) \displaystyle\hat{f}(\omega)
  443. f ^ ( ξ ) \displaystyle\hat{f}(\xi)
  444. f ^ ( ν ) \displaystyle\hat{f}(\nu)
  445. f ( x ) ¯ \displaystyle\overline{f(x)}
  446. f ^ ( - ξ ) ¯ \displaystyle\overline{\hat{f}(-\xi)}
  447. f ^ ( - ω ) ¯ \displaystyle\overline{\hat{f}(-\omega)}
  448. f ^ ( - ν ) ¯ \displaystyle\overline{\hat{f}(-\nu)}
  449. f ( x ) \displaystyle f(x)
  450. f ^ ( ξ ) = \displaystyle\hat{f}(\xi)=
  451. - f ( x ) e - 2 π i x ξ d x \displaystyle\int_{-\infty}^{\infty}f(x)e^{-2\pi ix\xi}\,dx
  452. f ^ ( ω ) = \displaystyle\hat{f}(\omega)=
  453. 1 2 π - f ( x ) e - i ω x d x \displaystyle\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(x)e^{-i\omega x}\,dx
  454. f ^ ( ν ) = \displaystyle\hat{f}(\nu)=
  455. - f ( x ) e - i ν x d x \displaystyle\int_{-\infty}^{\infty}f(x)e^{-i\nu x}\,dx
  456. rect ( a x ) \displaystyle\operatorname{rect}(ax)\,
  457. 1 | a | sinc ( ξ a ) \displaystyle\frac{1}{|a|}\cdot\operatorname{sinc}\left(\frac{\xi}{a}\right)
  458. 1 2 π a 2 sinc ( ω 2 π a ) \displaystyle\frac{1}{\sqrt{2\pi a^{2}}}\cdot\operatorname{sinc}\left(\frac{% \omega}{2\pi a}\right)
  459. 1 | a | sinc ( ν 2 π a ) \displaystyle\frac{1}{|a|}\cdot\operatorname{sinc}\left(\frac{\nu}{2\pi a}\right)
  460. sinc ( a x ) \displaystyle\operatorname{sinc}(ax)\,
  461. 1 | a | rect ( ξ a ) \displaystyle\frac{1}{|a|}\cdot\operatorname{rect}\left(\frac{\xi}{a}\right)\,
  462. 1 2 π a 2 rect ( ω 2 π a ) \displaystyle\frac{1}{\sqrt{2\pi a^{2}}}\cdot\operatorname{rect}\left(\frac{% \omega}{2\pi a}\right)
  463. 1 | a | rect ( ν 2 π a ) \displaystyle\frac{1}{|a|}\cdot\operatorname{rect}\left(\frac{\nu}{2\pi a}\right)
  464. sinc 2 ( a x ) \displaystyle\operatorname{sinc}^{2}(ax)
  465. 1 | a | tri ( ξ a ) \displaystyle\frac{1}{|a|}\cdot\operatorname{tri}\left(\frac{\xi}{a}\right)
  466. 1 2 π a 2 tri ( ω 2 π a ) \displaystyle\frac{1}{\sqrt{2\pi a^{2}}}\cdot\operatorname{tri}\left(\frac{% \omega}{2\pi a}\right)
  467. 1 | a | tri ( ν 2 π a ) \displaystyle\frac{1}{|a|}\cdot\operatorname{tri}\left(\frac{\nu}{2\pi a}\right)
  468. tri ( a x ) \displaystyle\operatorname{tri}(ax)
  469. 1 | a | sinc 2 ( ξ a ) \displaystyle\frac{1}{|a|}\cdot\operatorname{sinc}^{2}\left(\frac{\xi}{a}% \right)\,
  470. 1 2 π a 2 sinc 2 ( ω 2 π a ) \displaystyle\frac{1}{\sqrt{2\pi a^{2}}}\cdot\operatorname{sinc}^{2}\left(% \frac{\omega}{2\pi a}\right)
  471. 1 | a | sinc 2 ( ν 2 π a ) \displaystyle\frac{1}{|a|}\cdot\operatorname{sinc}^{2}\left(\frac{\nu}{2\pi a}\right)
  472. e - a x u ( x ) \displaystyle e^{-ax}u(x)\,
  473. 1 a + 2 π i ξ \displaystyle\frac{1}{a+2\pi i\xi}
  474. 1 2 π ( a + i ω ) \displaystyle\frac{1}{\sqrt{2\pi}(a+i\omega)}
  475. 1 a + i ν \displaystyle\frac{1}{a+i\nu}
  476. e - α x 2 \displaystyle e^{-\alpha x^{2}}\,
  477. π α e - ( π ξ ) 2 α \displaystyle\sqrt{\frac{\pi}{\alpha}}\cdot e^{-\frac{(\pi\xi)^{2}}{\alpha}}
  478. 1 2 α e - ω 2 4 α \displaystyle\frac{1}{\sqrt{2\alpha}}\cdot e^{-\frac{\omega^{2}}{4\alpha}}
  479. π α e - ν 2 4 α \displaystyle\sqrt{\frac{\pi}{\alpha}}\cdot e^{-\frac{\nu^{2}}{4\alpha}}
  480. e - a | x | \displaystyle\operatorname{e}^{-a|x|}\,
  481. 2 a a 2 + 4 π 2 ξ 2 \displaystyle\frac{2a}{a^{2}+4\pi^{2}\xi^{2}}
  482. 2 π a a 2 + ω 2 \displaystyle\sqrt{\frac{2}{\pi}}\cdot\frac{a}{a^{2}+\omega^{2}}
  483. 2 a a 2 + ν 2 \displaystyle\frac{2a}{a^{2}+\nu^{2}}
  484. sech ( a x ) \displaystyle\operatorname{sech}(ax)\,
  485. π a sech ( π 2 a ξ ) \displaystyle\frac{\pi}{a}\operatorname{sech}\left(\frac{\pi^{2}}{a}\xi\right)
  486. 1 a π 2 sech ( π 2 a ω ) \displaystyle\frac{1}{a}\sqrt{\frac{\pi}{2}}\operatorname{sech}\left(\frac{\pi% }{2a}\omega\right)
  487. π a sech ( π 2 a ν ) \displaystyle\frac{\pi}{a}\operatorname{sech}\left(\frac{\pi}{2a}\nu\right)
  488. e - a 2 x 2 2 H n ( a x ) \displaystyle e^{-\frac{a^{2}x^{2}}{2}}H_{n}(ax)\,
  489. 2 π ( - i ) n a \displaystyle\frac{\sqrt{2\pi}(-i)^{n}}{a}
  490. e - 2 π 2 ξ 2 a 2 H n ( 2 π ξ a ) \cdot e^{-\frac{2\pi^{2}\xi^{2}}{a^{2}}}H_{n}\left(\frac{2\pi\xi}{a}\right)
  491. ( - i ) n a \displaystyle\frac{(-i)^{n}}{a}
  492. e - ω 2 2 a 2 H n ( ω a ) \cdot e^{-\frac{\omega^{2}}{2a^{2}}}H_{n}\left(\frac{\omega}{a}\right)
  493. ( - i ) n 2 π a \displaystyle\frac{(-i)^{n}\sqrt{2\pi}}{a}
  494. e - ν 2 2 a 2 H n ( ν a ) \cdot e^{-\frac{\nu^{2}}{2a^{2}}}H_{n}\left(\frac{\nu}{a}\right)
  495. H n H_{n}
  496. f ( x ) \displaystyle f(x)
  497. f ^ ( ξ ) = \displaystyle\hat{f}(\xi)=
  498. - f ( x ) e - 2 π i x ξ d x \displaystyle\int_{-\infty}^{\infty}f(x)e^{-2\pi ix\xi}\,dx
  499. f ^ ( ω ) = \displaystyle\hat{f}(\omega)=
  500. 1 2 π - f ( x ) e - i ω x d x \displaystyle\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(x)e^{-i\omega x}\,dx
  501. f ^ ( ν ) = \displaystyle\hat{f}(\nu)=
  502. - f ( x ) e - i ν x d x \displaystyle\int_{-\infty}^{\infty}f(x)e^{-i\nu x}\,dx
  503. 1 \displaystyle 1
  504. δ ( ξ ) \displaystyle\delta(\xi)
  505. 2 π δ ( ω ) \displaystyle\sqrt{2\pi}\cdot\delta(\omega)
  506. 2 π δ ( ν ) \displaystyle 2\pi\delta(\nu)
  507. δ ( x ) \displaystyle\delta(x)\,
  508. 1 \displaystyle 1
  509. 1 2 π \displaystyle\frac{1}{\sqrt{2\pi}}\,
  510. 1 \displaystyle 1
  511. e i a x \displaystyle e^{iax}
  512. δ ( ξ - a 2 π ) \displaystyle\delta\left(\xi-\frac{a}{2\pi}\right)
  513. 2 π δ ( ω - a ) \displaystyle\sqrt{2\pi}\cdot\delta(\omega-a)
  514. 2 π δ ( ν - a ) \displaystyle 2\pi\delta(\nu-a)
  515. cos ( a x ) \displaystyle\cos(ax)
  516. δ ( ξ - a 2 π ) + δ ( ξ + a 2 π ) 2 \displaystyle\frac{\displaystyle\delta\left(\xi-\frac{a}{2\pi}\right)+\delta% \left(\xi+\frac{a}{2\pi}\right)}{2}
  517. 2 π δ ( ω - a ) + δ ( ω + a ) 2 \displaystyle\sqrt{2\pi}\cdot\frac{\delta(\omega-a)+\delta(\omega+a)}{2}\,
  518. π ( δ ( ν - a ) + δ ( ν + a ) ) \displaystyle\pi\left(\delta(\nu-a)+\delta(\nu+a)\right)
  519. cos ( a x ) = \textstyle\cos(ax)=
  520. ( e i a x + e - i a x ) / 2. (e^{iax}+e^{-iax})/2.
  521. sin ( a x ) \displaystyle\sin(ax)
  522. δ ( ξ - a 2 π ) - δ ( ξ + a 2 π ) 2 i \displaystyle\frac{\displaystyle\delta\left(\xi-\frac{a}{2\pi}\right)-\delta% \left(\xi+\frac{a}{2\pi}\right)}{2i}
  523. 2 π δ ( ω - a ) - δ ( ω + a ) 2 i \displaystyle\sqrt{2\pi}\cdot\frac{\delta(\omega-a)-\delta(\omega+a)}{2i}
  524. - i π ( δ ( ν - a ) - δ ( ν + a ) ) \displaystyle-i\pi\left(\delta(\nu-a)-\delta(\nu+a)\right)
  525. sin ( a x ) = \textstyle\sin(ax)=
  526. ( e i a x - e - i a x ) / ( 2 i ) . (e^{iax}-e^{-iax})/(2i).
  527. cos ( a x 2 ) \displaystyle\cos(ax^{2})
  528. π a cos ( π 2 ξ 2 a - π 4 ) \displaystyle\sqrt{\frac{\pi}{a}}\cos\left(\frac{\pi^{2}\xi^{2}}{a}-\frac{\pi}% {4}\right)
  529. 1 2 a cos ( ω 2 4 a - π 4 ) \displaystyle\frac{1}{\sqrt{2a}}\cos\left(\frac{\omega^{2}}{4a}-\frac{\pi}{4}\right)
  530. π a cos ( ν 2 4 a - π 4 ) \displaystyle\sqrt{\frac{\pi}{a}}\cos\left(\frac{\nu^{2}}{4a}-\frac{\pi}{4}\right)
  531. sin ( a x 2 ) \displaystyle\sin(ax^{2})\,
  532. - π a sin ( π 2 ξ 2 a - π 4 ) \displaystyle-\sqrt{\frac{\pi}{a}}\sin\left(\frac{\pi^{2}\xi^{2}}{a}-\frac{\pi% }{4}\right)
  533. - 1 2 a sin ( ω 2 4 a - π 4 ) \displaystyle\frac{-1}{\sqrt{2a}}\sin\left(\frac{\omega^{2}}{4a}-\frac{\pi}{4}\right)
  534. - π a sin ( ν 2 4 a - π 4 ) \displaystyle-\sqrt{\frac{\pi}{a}}\sin\left(\frac{\nu^{2}}{4a}-\frac{\pi}{4}\right)
  535. x n \displaystyle x^{n}\,
  536. ( i 2 π ) n δ ( n ) ( ξ ) \displaystyle\left(\frac{i}{2\pi}\right)^{n}\delta^{(n)}(\xi)\,
  537. i n 2 π δ ( n ) ( ω ) \displaystyle i^{n}\sqrt{2\pi}\delta^{(n)}(\omega)\,
  538. 2 π i n δ ( n ) ( ν ) \displaystyle 2\pi i^{n}\delta^{(n)}(\nu)\,
  539. δ ( n ) ( ξ ) \textstyle\delta^{(n)}(\xi)
  540. 1 x \displaystyle\frac{1}{x}
  541. - i π sgn ( ξ ) \displaystyle-i\pi\operatorname{sgn}(\xi)
  542. - i π 2 sgn ( ω ) \displaystyle-i\sqrt{\frac{\pi}{2}}\operatorname{sgn}(\omega)
  543. - i π sgn ( ν ) \displaystyle-i\pi\operatorname{sgn}(\nu)
  544. 1 x n := \displaystyle\frac{1}{x^{n}}:=
  545. ( - 1 ) n - 1 ( n - 1 ) ! d n d x n log | x | \displaystyle\frac{(-1)^{n-1}}{(n-1)!}\frac{d^{n}}{dx^{n}}\log|x|
  546. - i π ( - 2 π i ξ ) n - 1 ( n - 1 ) ! sgn ( ξ ) \displaystyle-i\pi\frac{(-2\pi i\xi)^{n-1}}{(n-1)!}\operatorname{sgn}(\xi)
  547. - i π 2 ( - i ω ) n - 1 ( n - 1 ) ! sgn ( ω ) \displaystyle-i\sqrt{\frac{\pi}{2}}\cdot\frac{(-i\omega)^{n-1}}{(n-1)!}% \operatorname{sgn}(\omega)
  548. - i π ( - i ν ) n - 1 ( n - 1 ) ! sgn ( ν ) \displaystyle-i\pi\frac{(-i\nu)^{n-1}}{(n-1)!}\operatorname{sgn}(\nu)
  549. ( - 1 ) n - 1 ( n - 1 ) ! d n d x n log | x | \textstyle\frac{(-1)^{n-1}}{(n-1)!}\frac{d^{n}}{dx^{n}}\log|x|
  550. | x | α \displaystyle|x|^{\alpha}\,
  551. - 2 sin ( π α / 2 ) Γ ( α + 1 ) | 2 π ξ | α + 1 \displaystyle-2\frac{\sin(\pi\alpha/2)\Gamma(\alpha+1)}{|2\pi\xi|^{\alpha+1}}
  552. - 2 2 π sin ( π α / 2 ) Γ ( α + 1 ) | ω | α + 1 \displaystyle\frac{-2}{\sqrt{2\pi}}\frac{\sin(\pi\alpha/2)\Gamma(\alpha+1)}{|% \omega|^{\alpha+1}}
  553. - 2 sin ( π α / 2 ) Γ ( α + 1 ) | ν | α + 1 \displaystyle-2\frac{\sin(\pi\alpha/2)\Gamma(\alpha+1)}{|\nu|^{\alpha+1}}
  554. | x | α |x|^{\alpha}
  555. α | x | α \textstyle\alpha\mapsto|x|^{\alpha}
  556. | x | α |x|^{\alpha}
  557. 1 | x | \frac{1}{\sqrt{|x|}}\,
  558. 1 | ξ | \frac{1}{\sqrt{|\xi|}}
  559. 1 | ω | \frac{1}{\sqrt{|\omega|}}
  560. 2 π | ν | \frac{\sqrt{2\pi}}{\sqrt{|\nu|}}
  561. sgn ( x ) \displaystyle\operatorname{sgn}(x)
  562. 1 i π ξ \displaystyle\frac{1}{i\pi\xi}
  563. 2 π 1 i ω \displaystyle\sqrt{\frac{2}{\pi}}\frac{1}{i\omega}
  564. 2 i ν \displaystyle\frac{2}{i\nu}
  565. u ( x ) \displaystyle u(x)
  566. 1 2 ( 1 i π ξ + δ ( ξ ) ) \displaystyle\frac{1}{2}\left(\frac{1}{i\pi\xi}+\delta(\xi)\right)
  567. π 2 ( 1 i π ω + δ ( ω ) ) \displaystyle\sqrt{\frac{\pi}{2}}\left(\frac{1}{i\pi\omega}+\delta(\omega)\right)
  568. π ( 1 i π ν + δ ( ν ) ) \displaystyle\pi\left(\frac{1}{i\pi\nu}+\delta(\nu)\right)
  569. n = - δ ( x - n T ) \displaystyle\sum_{n=-\infty}^{\infty}\delta(x-nT)
  570. 1 T k = - δ ( ξ - k T ) \displaystyle\frac{1}{T}\sum_{k=-\infty}^{\infty}\delta\left(\xi-\frac{k}{T}\right)
  571. 2 π T k = - δ ( ω - 2 π k T ) \displaystyle\frac{\sqrt{2\pi}}{T}\sum_{k=-\infty}^{\infty}\delta\left(\omega-% \frac{2\pi k}{T}\right)
  572. 2 π T k = - δ ( ν - 2 π k T ) \displaystyle\frac{2\pi}{T}\sum_{k=-\infty}^{\infty}\delta\left(\nu-\frac{2\pi k% }{T}\right)
  573. n = - e i n x = \sum_{n=-\infty}^{\infty}e^{inx}=
  574. 2 π k = - δ ( x + 2 π k ) 2\pi\sum_{k=-\infty}^{\infty}\delta(x+2\pi k)
  575. J 0 ( x ) \displaystyle J_{0}(x)
  576. 2 rect ( π ξ ) 1 - 4 π 2 ξ 2 \displaystyle\frac{2\,\operatorname{rect}(\pi\xi)}{\sqrt{1-4\pi^{2}\xi^{2}}}
  577. 2 π rect ( ω 2 ) 1 - ω 2 \displaystyle\sqrt{\frac{2}{\pi}}\cdot\frac{\operatorname{rect}\left(% \displaystyle\frac{\omega}{2}\right)}{\sqrt{1-\omega^{2}}}
  578. 2 rect ( ν 2 ) 1 - ν 2 \displaystyle\frac{2\,\operatorname{rect}\left(\displaystyle\frac{\nu}{2}% \right)}{\sqrt{1-\nu^{2}}}
  579. J n ( x ) \displaystyle J_{n}(x)
  580. 2 ( - i ) n T n ( 2 π ξ ) rect ( π ξ ) 1 - 4 π 2 ξ 2 \displaystyle\frac{2(-i)^{n}T_{n}(2\pi\xi)\operatorname{rect}(\pi\xi)}{\sqrt{1% -4\pi^{2}\xi^{2}}}
  581. 2 π ( - i ) n T n ( ω ) rect ( ω 2 ) 1 - ω 2 \displaystyle\sqrt{\frac{2}{\pi}}\frac{(-i)^{n}T_{n}(\omega)\operatorname{rect% }\left(\displaystyle\frac{\omega}{2}\right)}{\sqrt{1-\omega^{2}}}
  582. 2 ( - i ) n T n ( ν ) rect ( ν 2 ) 1 - ν 2 \displaystyle\frac{2(-i)^{n}T_{n}(\nu)\operatorname{rect}\left(\displaystyle% \frac{\nu}{2}\right)}{\sqrt{1-\nu^{2}}}
  583. log | x | \displaystyle\log\left|x\right|
  584. - 1 2 1 | ξ | - γ δ ( ξ ) \displaystyle-\frac{1}{2}\frac{1}{\left|\xi\right|}-\gamma\delta\left(\xi\right)
  585. - π / 2 | ω | - 2 π γ δ ( ω ) \displaystyle-\frac{\sqrt{\pi/2}}{\left|\omega\right|}-\sqrt{2\pi}\gamma\delta% \left(\omega\right)
  586. - π | ν | - 2 π γ δ ( ν ) \displaystyle-\frac{\pi}{\left|\nu\right|}-2\pi\gamma\delta\left(\nu\right)
  587. γ \gamma
  588. ( i x ) - α \displaystyle\left(\mp ix\right)^{-\alpha}
  589. ( 2 π ) α Γ ( α ) u ( ± ξ ) ( ± ξ ) α - 1 \displaystyle\frac{\left(2\pi\right)^{\alpha}}{\Gamma\left(\alpha\right)}u% \left(\pm\xi\right)\left(\pm\xi\right)^{\alpha-1}
  590. 2 π Γ ( α ) u ( ± ω ) ( ± ω ) α - 1 \displaystyle\frac{\sqrt{2\pi}}{\Gamma\left(\alpha\right)}u\left(\pm\omega% \right)\left(\pm\omega\right)^{\alpha-1}
  591. 2 π Γ ( α ) u ( ± ν ) ( ± ν ) α - 1 \displaystyle\frac{2\pi}{\Gamma\left(\alpha\right)}u\left(\pm\nu\right)\left(% \pm\nu\right)^{\alpha-1}
  592. f ( x , y ) \displaystyle f(x,y)
  593. f ^ ( ξ x , ξ y ) = \displaystyle\hat{f}(\xi_{x},\xi_{y})=
  594. f ( x , y ) e - 2 π i ( ξ x x + ξ y y ) d x d y \displaystyle\iint f(x,y)e^{-2\pi i(\xi_{x}x+\xi_{y}y)}\,dx\,dy
  595. f ^ ( ω x , ω y ) = \displaystyle\hat{f}(\omega_{x},\omega_{y})=
  596. 1 2 π f ( x , y ) e - i ( ω x x + ω y y ) d x d y \displaystyle\frac{1}{2\pi}\iint f(x,y)e^{-i(\omega_{x}x+\omega_{y}y)}\,dx\,dy
  597. f ^ ( ν x , ν y ) = \displaystyle\hat{f}(\nu_{x},\nu_{y})=
  598. f ( x , y ) e - i ( ν x x + ν y y ) d x d y \displaystyle\iint f(x,y)e^{-i(\nu_{x}x+\nu_{y}y)}\,dx\,dy
  599. e - π ( a 2 x 2 + b 2 y 2 ) \displaystyle e^{-\pi\left(a^{2}x^{2}+b^{2}y^{2}\right)}
  600. 1 | a b | e - π ( ξ x 2 / a 2 + ξ y 2 / b 2 ) \displaystyle\frac{1}{|ab|}e^{-\pi\left(\xi_{x}^{2}/a^{2}+\xi_{y}^{2}/b^{2}% \right)}
  601. 1 2 π | a b | e - ( ω x 2 / a 2 + ω y 2 / b 2 ) 4 π \displaystyle\frac{1}{2\pi\cdot|ab|}e^{\frac{-\left(\omega_{x}^{2}/a^{2}+% \omega_{y}^{2}/b^{2}\right)}{4\pi}}
  602. 1 | a b | e - ( ν x 2 / a 2 + ν y 2 / b 2 ) 4 π \displaystyle\frac{1}{|ab|}e^{\frac{-\left(\nu_{x}^{2}/a^{2}+\nu_{y}^{2}/b^{2}% \right)}{4\pi}}
  603. circ ( x 2 + y 2 ) \displaystyle\mathrm{circ}(\sqrt{x^{2}+y^{2}})
  604. J 1 ( 2 π ξ x 2 + ξ y 2 ) ξ x 2 + ξ y 2 \displaystyle\frac{J_{1}\left(2\pi\sqrt{\xi_{x}^{2}+\xi_{y}^{2}}\right)}{\sqrt% {\xi_{x}^{2}+\xi_{y}^{2}}}
  605. J 1 ( ω x 2 + ω y 2 ) ω x 2 + ω y 2 \displaystyle\frac{J_{1}\left(\sqrt{\omega_{x}^{2}+\omega_{y}^{2}}\right)}{% \sqrt{\omega_{x}^{2}+\omega_{y}^{2}}}
  606. 2 π J 1 ( ν x 2 + ν y 2 ) ν x 2 + ν y 2 \displaystyle\frac{2\pi J_{1}\left(\sqrt{\nu_{x}^{2}+\nu_{y}^{2}}\right)}{% \sqrt{\nu_{x}^{2}+\nu_{y}^{2}}}
  607. f ( 𝐱 ) \displaystyle f(\mathbf{x})\,
  608. f ^ ( s y m b o l ξ ) = \displaystyle\hat{f}(symbol\xi)=
  609. 𝐑 n f ( 𝐱 ) e - 2 π i 𝐱 s y m b o l ξ d n 𝐱 \displaystyle\int_{\mathbf{R}^{n}}f(\mathbf{x})e^{-2\pi i\mathbf{x}\cdot symbol% \xi}\,d^{n}\mathbf{x}
  610. f ^ ( s y m b o l ω ) = \displaystyle\hat{f}(symbol\omega)=
  611. 1 ( 2 π ) n / 2 𝐑 n f ( 𝐱 ) e - i s y m b o l ω 𝐱 d n 𝐱 \displaystyle\frac{1}{{(2\pi)}^{n/2}}\int_{\mathbf{R}^{n}}f(\mathbf{x})e^{-% isymbol\omega\cdot\mathbf{x}}\,d^{n}\mathbf{x}
  612. f ^ ( s y m b o l ν ) = \displaystyle\hat{f}(symbol\nu)=
  613. 𝐑 n f ( 𝐱 ) e - i 𝐱 s y m b o l ν d n 𝐱 \displaystyle\int_{\mathbf{R}^{n}}f(\mathbf{x})e^{-i\mathbf{x}\cdot symbol\nu}% \,d^{n}\mathbf{x}
  614. χ [ 0 , 1 ] ( | 𝐱 | ) ( 1 - | 𝐱 | 2 ) δ \displaystyle\chi_{[0,1]}(|\mathbf{x}|)(1-|\mathbf{x}|^{2})^{\delta}
  615. π - δ Γ ( δ + 1 ) | s y m b o l ξ | - n / 2 - δ \displaystyle\pi^{-\delta}\Gamma(\delta+1)|symbol\xi|^{-n/2-\delta}
  616. × J n / 2 + δ ( 2 π | s y m b o l ξ | ) \displaystyle\times J_{n/2+\delta}(2\pi|symbol\xi|)
  617. 2 - δ Γ ( δ + 1 ) | s y m b o l ω | - n / 2 - δ \displaystyle 2^{-\delta}\Gamma(\delta+1)\left|symbol\omega\right|^{-n/2-\delta}
  618. × J n / 2 + δ ( | s y m b o l ω | ) \displaystyle\times J_{n/2+\delta}(|symbol\omega|)
  619. π - δ Γ ( δ + 1 ) | s y m b o l ν 2 π | - n / 2 - δ \displaystyle\pi^{-\delta}\Gamma(\delta+1)\left|\frac{symbol\nu}{2\pi}\right|^% {-n/2-\delta}
  620. × J n / 2 + δ ( | s y m b o l ν | ) \displaystyle\times J_{n/2+\delta}(|symbol\nu|)
  621. | 𝐱 | - α , 0 < Re α < n . \displaystyle|\mathbf{x}|^{-\alpha},\quad 0<\operatorname{Re}\alpha<n.
  622. c n - α , n | s y m b o l ξ | - ( n - α ) \displaystyle c_{n-\alpha,n}|symbol\xi|^{-(n-\alpha)}
  623. 1 s y m b o l σ ( 2 π ) n / 2 e - 1 2 𝐱 T s y m b o l σ - T s y m b o l σ - 1 𝐱 \displaystyle\frac{1}{\left\|symbol\sigma\right\|\left(2\pi\right)^{n/2}}e^{-% \frac{1}{2}\mathbf{x}^{\mathrm{T}}symbol\sigma^{-\mathrm{T}}symbol\sigma^{-1}% \mathbf{x}}
  624. e - 1 2 s y m b o l ν T s y m b o l σ s y m b o l σ T s y m b o l ν \displaystyle e^{-\frac{1}{2}symbol\nu^{\mathrm{T}}symbol\sigma symbol\sigma^{% \mathrm{T}}symbol\nu}
  625. e - 2 π α | 𝐱 | \displaystyle e^{-2\pi\alpha|\mathbf{x}|}
  626. c n α ( α 2 + | ξ | 2 ) ( n + 1 ) / 2 c_{n}\frac{\alpha}{(\alpha^{2}+|\xi|^{2})^{(n+1)/2}}
  627. c α , n = π n / 2 2 α Γ ( α / 2 ) Γ ( ( n - α ) / 2 ) c_{\alpha,n}=\pi^{n/2}2^{\alpha}\frac{\Gamma(\alpha/2)}{\Gamma((n-\alpha)/2)}
  628. s y m b o l Σ = s y m b o l σ s y m b o l σ T symbol\Sigma=symbol\sigma symbol\sigma^{\mathrm{T}}
  629. s y m b o l Σ - 1 = s y m b o l σ - T s y m b o l σ - 1 symbol\Sigma^{-1}=symbol\sigma^{-\mathrm{T}}symbol\sigma^{-1}
  630. c n = Γ ( ( n + 1 ) / 2 ) / π ( n + 1 ) / 2 c_{n}=\Gamma((n+1)/2)/\pi^{(n+1)/2}

Fourier_transform_spectroscopy.html

  1. p p
  2. ν ~ = 1 / λ \tilde{\nu}=1/\lambda
  3. I ( p , ν ~ ) = I ( ν ~ ) [ 1 + cos ( 2 π ν ~ p ) ] I(p,\tilde{\nu})=I(\tilde{\nu})[1+\cos(2\pi\tilde{\nu}p)]
  4. I ( ν ~ ) I(\tilde{\nu})
  5. I ( ν ~ ) I(\tilde{\nu})
  6. I ( p ) = ( p , 0 I ( p , ν ~ ) d ν ~ ) = ( p , 0 I ( ν ~ ) [ 1 + cos ( 2 π ν ~ p ) ] d ν ~ ) . I(p)=(p,\int_{0}^{\infty}I(p,\tilde{\nu})d\tilde{\nu})=(p,\int_{0}^{\infty}I(% \tilde{\nu})[1+\cos(2\pi\tilde{\nu}p)]d\tilde{\nu}).
  7. I ( p ) I(p)
  8. I ( ν ~ ) = 4 0 [ I ( p ) - 1 2 I ( p = 0 ) ] cos ( 2 π ν ~ p ) d p . I(\tilde{\nu})=4\int_{0}^{\infty}[I(p)-\tfrac{1}{2}I(p=0)]\cos(2\pi\tilde{\nu}% p)dp.

Fractal_art.html

  1. z 2 + a z 4 + c z^{2}+az^{4}+c

Fractal_compression.html

  1. 2 \mathbb{R}^{2}
  2. f i : 2 2 . f_{i}:\mathbb{R}^{2}\to\mathbb{R}^{2}.
  3. H ( A ) = i = 1 N f i ( A ) , A 2 . H(A)=\bigcup_{i=1}^{N}f_{i}(A),\quad A\subset\mathbb{R}^{2}.
  4. S = f 1 ( S ) f 2 ( S ) f N ( S ) S=f_{1}(S)\cup f_{2}(S)\cup\cdots\cup f_{N}(S)
  5. 3 \mathbb{R}^{3}
  6. 3 \mathbb{R}^{3}
  7. f i : 3 3 . f_{i}:\mathbb{R}^{3}\to\mathbb{R}^{3}.

Frame_problem.html

  1. 𝑜𝑝𝑒𝑛 \,\textit{open}
  2. 𝑜𝑛 \,\textit{on}
  3. 𝑜𝑝𝑒𝑛 ( t ) \,\textit{open}(t)
  4. 𝑜𝑛 ( t ) \,\textit{on}(t)
  5. ¬ 𝑜𝑝𝑒𝑛 ( 0 ) \neg\,\textit{open}(0)
  6. ¬ 𝑜𝑛 ( 0 ) \neg\,\textit{on}(0)
  7. 𝑡𝑟𝑢𝑒 𝑜𝑝𝑒𝑛 ( 1 ) \,\textit{true}\rightarrow\,\textit{open}(1)
  8. ¬ 𝑙𝑜𝑐𝑘𝑒𝑑 ( 0 ) 𝑜𝑝𝑒𝑛 ( 1 ) \neg\,\textit{locked}(0)\rightarrow\,\textit{open}(1)
  9. 𝑒𝑥𝑒𝑐𝑢𝑡𝑒𝑜𝑝𝑒𝑛 ( t ) \,\textit{executeopen}(t)
  10. t . 𝑒𝑥𝑒𝑐𝑢𝑡𝑒𝑜𝑝𝑒𝑛 ( t ) 𝑡𝑟𝑢𝑒 𝑜𝑝𝑒𝑛 ( t + 1 ) \forall t.\,\textit{executeopen}(t)\wedge\,\textit{true}\rightarrow\,\textit{% open}(t+1)
  11. ¬ 𝑜𝑝𝑒𝑛 ( 0 ) \neg\,\textit{open}(0)
  12. 𝑜𝑝𝑒𝑛 ( 1 ) \,\textit{open}(1)
  13. ¬ 𝑜𝑛 ( 0 ) \neg\,\textit{on}(0)
  14. ¬ 𝑜𝑛 ( 1 ) \neg\,\textit{on}(1)
  15. ¬ 𝑜𝑝𝑒𝑛 ( 0 ) \neg\,\textit{open}(0)
  16. 𝑜𝑝𝑒𝑛 ( 1 ) \,\textit{open}(1)
  17. ¬ 𝑜𝑛 ( 0 ) \neg\,\textit{on}(0)
  18. 𝑜𝑛 ( 1 ) \,\textit{on}(1)
  19. 𝑜𝑛 ( 0 ) 𝑜𝑛 ( 1 ) \,\textit{on}(0)\leftrightarrow\,\textit{on}(1)
  20. 𝑜𝑐𝑐𝑙𝑢𝑑𝑒𝑜𝑝𝑒𝑛 ( t ) \,\textit{occludeopen}(t)
  21. 𝑜𝑐𝑐𝑙𝑢𝑑𝑒𝑜𝑛 ( t ) \,\textit{occludeon}(t)
  22. ¬ 𝑜𝑝𝑒𝑛 ( 0 ) \neg\,\textit{open}(0)
  23. ¬ 𝑜𝑛 ( 0 ) \neg\,\textit{on}(0)
  24. 𝑡𝑟𝑢𝑒 𝑜𝑝𝑒𝑛 ( 1 ) 𝑜𝑐𝑐𝑙𝑢𝑑𝑒𝑜𝑝𝑒𝑛 ( 1 ) \,\textit{true}\rightarrow\,\textit{open}(1)\wedge\,\textit{occludeopen}(1)
  25. t . ¬ 𝑜𝑐𝑐𝑙𝑢𝑑𝑒𝑜𝑝𝑒𝑛 ( t ) ( 𝑜𝑝𝑒𝑛 ( t - 1 ) 𝑜𝑝𝑒𝑛 ( t ) ) \forall t.\neg\,\textit{occludeopen}(t)\rightarrow(\,\textit{open}(t-1)% \leftrightarrow\,\textit{open}(t))
  26. t . ¬ 𝑜𝑐𝑐𝑙𝑢𝑑𝑒𝑜𝑛 ( t ) ( 𝑜𝑛 ( t - 1 ) 𝑜𝑛 ( t ) ) \forall t.\neg\,\textit{occludeon}(t)\rightarrow(\,\textit{on}(t-1)% \leftrightarrow\,\textit{on}(t))
  27. 𝑜𝑐𝑐𝑙𝑢𝑑𝑒𝑜𝑝𝑒𝑛 ( 1 ) \,\textit{occludeopen}(1)
  28. t = 1 t=1
  29. 𝑜𝑝𝑒𝑛 ( t - 1 ) 𝑜𝑝𝑒𝑛 ( t ) \,\textit{open}(t-1)\leftrightarrow\,\textit{open}(t)
  30. t = 1 t=1
  31. 𝑜𝑝𝑒𝑛 \,\textit{open}
  32. 𝑜𝑐𝑐𝑙𝑢𝑑𝑒𝑜𝑝𝑒𝑛 \,\textit{occludeopen}
  33. 𝑜𝑝𝑒𝑛 \,\textit{open}
  34. 𝑜𝑝𝑒𝑛 \,\textit{open}
  35. 𝑐ℎ𝑎𝑛𝑔𝑒𝑜𝑝𝑒𝑛 ( t ) \,\textit{changeopen}(t)
  36. 𝑜𝑝𝑒𝑛 \,\textit{open}
  37. t t
  38. t + 1 t+1
  39. ¬ 𝑜𝑝𝑒𝑛 ( 0 ) \neg\,\textit{open}(0)
  40. ¬ 𝑜𝑛 ( 0 ) \neg\,\textit{on}(0)
  41. ¬ 𝑜𝑝𝑒𝑛 ( 0 ) 𝑡𝑟𝑢𝑒 𝑐ℎ𝑎𝑛𝑔𝑒𝑜𝑝𝑒𝑛 ( 0 ) \neg\,\textit{open}(0)\wedge\,\textit{true}\rightarrow\,\textit{changeopen}(0)
  42. t . 𝑐ℎ𝑎𝑛𝑔𝑒𝑜𝑝𝑒𝑛 ( t ) ( ¬ 𝑜𝑝𝑒𝑛 ( t ) 𝑜𝑝𝑒𝑛 ( t + 1 ) ) \forall t.\,\textit{changeopen}(t)\leftrightarrow(\neg\,\textit{open}(t)% \leftrightarrow\,\textit{open}(t+1))
  43. t . 𝑐ℎ𝑎𝑛𝑔𝑒𝑜𝑛 ( t ) ( ¬ 𝑜𝑛 ( t ) 𝑜𝑛 ( t + 1 ) ) \forall t.\,\textit{changeon}(t)\leftrightarrow(\neg\,\textit{on}(t)% \leftrightarrow\,\textit{on}(t+1))
  44. t t
  45. t t
  46. 𝑜𝑝𝑒𝑛𝑑𝑜𝑜𝑟 ( t ) \,\textit{opendoor}(t)
  47. 𝑐𝑙𝑜𝑠𝑒𝑑𝑜𝑜𝑟 ( t ) \,\textit{closedoor}(t)
  48. t t
  49. ¬ 𝑜𝑝𝑒𝑛 ( 0 ) \neg\,\textit{open}(0)
  50. ¬ 𝑜𝑛 ( 0 ) \neg\,\textit{on}(0)
  51. 𝑜𝑝𝑒𝑛𝑑𝑜𝑜𝑟 ( 0 ) \,\textit{opendoor}(0)
  52. t . 𝑜𝑝𝑒𝑛 ( t + 1 ) 𝑜𝑝𝑒𝑛𝑑𝑜𝑜𝑟 ( t ) ( 𝑜𝑝𝑒𝑛 ( t ) ¬ 𝑐𝑙𝑜𝑠𝑒𝑑𝑜𝑜𝑟 ( t ) ) \forall t.\,\textit{open}(t+1)\leftrightarrow\,\textit{opendoor}(t)\vee(\,% \textit{open}(t)\wedge\neg\,\textit{closedoor}(t))
  53. 𝑜𝑝𝑒𝑛 𝑜𝑛 \,\textit{open}\circ\,\textit{on}
  54. 𝑜𝑝𝑒𝑛 𝑜𝑛 \,\textit{open}\circ\,\textit{on}
  55. 𝑠𝑡𝑎𝑡𝑒 ( 𝑜𝑝𝑒𝑛 𝑜𝑛 , 10 ) \,\textit{state}(\,\textit{open}\circ\,\textit{on},10)
  56. 10 10
  57. 𝑠𝑡𝑎𝑡𝑒 ( s 𝑜𝑝𝑒𝑛 , 1 ) 𝑠𝑡𝑎𝑡𝑒 ( s , 0 ) \,\textit{state}(s\circ\,\textit{open},1)\leftrightarrow\,\textit{state}(s,0)
  58. 𝑠𝑡𝑎𝑡𝑒 ( s , 1 ) 𝑠𝑡𝑎𝑡𝑒 ( s 𝑜𝑝𝑒𝑛 , 0 ) \,\textit{state}(s,1)\leftrightarrow\,\textit{state}(s\circ\,\textit{open},0)
  59. 𝑠𝑡𝑎𝑡𝑒 \,\textit{state}
  60. \circ
  61. 𝑠𝑡𝑎𝑡𝑒 ( 𝑜𝑝𝑒𝑛 s 𝑜𝑝𝑒𝑛 , t ) \,\textit{state}(\,\textit{open}\circ s\circ\,\textit{open},t)
  62. s s
  63. t t
  64. R ( x , s ) : R ( x , 𝑑𝑜 ( a , s ) ) R ( x , 𝑑𝑜 ( a , s ) ) \frac{R(x,s)\;:\ R(x,\,\textit{do}(a,s))}{R(x,\,\textit{do}(a,s))}
  65. R ( x ) R(x)
  66. s s
  67. R ( x ) R(x)
  68. a a
  69. R ( x ) R(x)
  70. r ( X , T + 1 ) r ( X , T ) , not r ( X , T + 1 ) r(X,T+1)\leftarrow r(X,T),\ \hbox{not }\sim r(X,T+1)
  71. r ( X ) r(X)
  72. T T
  73. r ( X ) r(X)
  74. T + 1 T+1
  75. r ( X ) r(X)
  76. 𝑜𝑝𝑒𝑛𝑑𝑜𝑜𝑟 \,\textit{opendoor}
  77. 𝑜𝑝𝑒𝑛𝑑𝑜𝑜𝑟 \,\textit{opendoor}
  78. 𝑜𝑝𝑒𝑛 \,\textit{open}
  79. ¬ 𝑙𝑜𝑐𝑘𝑒𝑑 \neg\,\textit{locked}

Free-space_path_loss.html

  1. FSPL = ( 4 π d λ ) 2 = ( 4 π d f c ) 2 \begin{aligned}\displaystyle\mbox{FSPL}&\displaystyle=\left(\frac{4\pi d}{% \lambda}\right)^{2}\\ &\displaystyle=\left(\frac{4\pi df}{c}\right)^{2}\end{aligned}
  2. λ \ \lambda
  3. f \ f
  4. d \ d
  5. c \ c
  6. FSPL(dB) = 10 log 10 ( ( 4 π c d f ) 2 ) = 20 log 10 ( 4 π c d f ) = 20 log 10 ( d ) + 20 log 10 ( f ) + 20 log 10 ( 4 π c ) = 20 log 10 ( d ) + 20 log 10 ( f ) - 147.55 \begin{aligned}\displaystyle\mbox{FSPL(dB)}&\displaystyle=10\log_{10}\left(% \left(\frac{4\pi}{c}df\right)^{2}\right)\\ &\displaystyle=20\log_{10}\left(\frac{4\pi}{c}df\right)\\ &\displaystyle=20\log_{10}(d)+20\log_{10}(f)+20\log_{10}\left(\frac{4\pi}{c}% \right)\\ &\displaystyle=20\log_{10}(d)+20\log_{10}(f)-147.55\end{aligned}
  7. f \ f
  8. d \ d
  9. FSPL(dB) = 20 log 10 ( d ) + 20 log 10 ( f ) + 92.45 \ \mbox{FSPL(dB)}~{}=20\log_{10}(d)+20\log_{10}(f)+92.45
  10. d , f \ d,f
  11. - 87.55 \ -87.55
  12. d , f \ d,f
  13. - 27.55 \ -27.55
  14. d , f \ d,f
  15. 32.45 \ 32.45
  16. S = P t 1 4 π d 2 \ S=P_{t}\frac{1}{4\pi d^{2}}
  17. S \ S
  18. d \ d
  19. P t \ P_{t}
  20. G r = 1 G_{r}=1
  21. F S P L = P t P r G t G r FSPL=\frac{P_{t}}{P_{r}}G_{t}G_{r}
  22. G r = 1 G_{r}=1
  23. A = G λ 2 4 π A=G\frac{\lambda^{2}}{4\pi}\,
  24. G r = 1 G_{r}=1
  25. P r \ P_{r}
  26. P r = S λ 2 4 π \ P_{r}=S\frac{\lambda^{2}}{4\pi}
  27. S \ S

Free_group.html

  1. a b 3 c - 1 c a - 1 c ab^{3}c^{-1}ca^{-1}c\,
  2. a b 3 c - 1 c a - 1 c a b 3 a - 1 c . ab^{3}c^{-1}ca^{-1}c\;\;\longrightarrow\;\;ab^{3}\,a^{-1}c.

Freeman_Dyson.html

  1. 𝒯 , \mathcal{T}\,,

Freenet.html

  1. O { [ l o g ( n ) ] 2 } O\left\{\left[log\left(n\right)\right]^{2}\right\}
  2. O ( log 2 n ) O(\log^{2}n)

Frequency.html

  1. ν \nu
  2. T = 1 f . T=\frac{1}{f}.
  3. y ( t ) = sin ( θ ( t ) ) = sin ( ω t ) = sin ( 2 π f t ) y(t)=\sin\left(\theta(t)\right)=\sin(\omega t)=\sin(2\mathrm{\pi}ft)
  4. d θ d t = ω = 2 π f \frac{\mathrm{d}\theta}{\mathrm{d}t}=\omega=2\mathrm{\pi}f
  5. y ( t ) = sin ( θ ( t , x ) ) = sin ( ω t + k x ) y(t)=\sin\left(\theta(t,x)\right)=\sin(\omega t+kx)
  6. d θ d x = k \frac{\mathrm{d}\theta}{\mathrm{d}x}=k
  7. f = v λ . f=\frac{v}{\lambda}.
  8. f = c λ . f=\frac{c}{\lambda}.
  9. f = 71 15 s 4.7 Hz f=\frac{71}{15\,\,\text{s}}\approx 4.7\,\,\text{Hz}
  10. c = f λ \displaystyle c=f\lambda

Frequency_modulation.html

  1. x m ( t ) x_{m}(t)
  2. x c ( t ) = A c cos ( 2 π f c t ) x_{c}(t)=A_{c}\cos(2\pi f_{c}t)\,
  3. y ( t ) = A c cos ( 2 π 0 t f ( τ ) d τ ) y(t)=A_{c}\cos\left(2\pi\int_{0}^{t}f(\tau)d\tau\right)
  4. = A c cos ( 2 π 0 t [ f c + f Δ x m ( τ ) ] d τ ) =A_{c}\cos\left(2\pi\int_{0}^{t}\left[f_{c}+f_{\Delta}x_{m}(\tau)\right]d\tau\right)
  5. = A c cos ( 2 π f c t + 2 π f Δ 0 t x m ( τ ) d τ ) =A_{c}\cos\left(2\pi f_{c}t+2\pi f_{\Delta}\int_{0}^{t}x_{m}(\tau)d\tau\right)
  6. f ( τ ) f(\tau)\,
  7. f Δ f_{\Delta}\,
  8. 0 t x m ( τ ) d τ = A m cos ( 2 π f m t ) 2 π f m \int_{0}^{t}x_{m}(\tau)d\tau=\frac{A_{m}\cos(2\pi f_{m}t)}{2\pi f_{m}}\,
  9. y ( t ) = A c cos ( 2 π f c t + f Δ f m cos ( 2 π f m t ) ) y(t)=A_{c}\cos\left(2\pi f_{c}t+\frac{f_{\Delta}}{f_{m}}\cos\left(2\pi f_{m}t% \right)\right)\,
  10. A m A_{m}\,
  11. f Δ f_{\Delta}\,
  12. h = Δ f f m = f Δ | x m ( t ) | f m h=\frac{\Delta{}f}{f_{m}}=\frac{f_{\Delta}|x_{m}(t)|}{f_{m}}
  13. f m f_{m}\,
  14. Δ f \Delta{}f\,
  15. h 1 h\ll 1
  16. 2 f m 2f_{m}\,
  17. T s T_{s}\,
  18. f m = 1 2 T s f_{m}=\frac{1}{2T_{s}}\,
  19. f c f_{c}\,
  20. f c + Δ f f_{c}+\Delta{}f
  21. f c - Δ f f_{c}-\Delta{}f
  22. h 1 h\gg 1
  23. 2 f Δ 2f_{\Delta}\,
  24. Δ f \Delta{}f\,
  25. f m f_{m}
  26. B T B_{T}\,
  27. B T = 2 ( Δ f + f m ) \ B_{T}=2(\Delta f+f_{m})\,
  28. Δ f \Delta f\,
  29. f ( t ) f(t)\,
  30. f c f_{c}\,

Frequency_modulation_synthesis.html

  1. m ( t ) = B sin ( ω m t ) m(t)=B\,\sin(\omega_{m}t)\,
  2. F M ( t ) = A sin ( 0 t ( ω c + B sin ( ω m τ ) ) d τ ) = A sin ( ω c t - B ω m ( cos ( ω m t ) - 1 ) ) = A sin ( ω c t + B ω m ( sin ( ω m t - π / 2 ) + 1 ) ) \begin{aligned}\displaystyle FM(t)&\displaystyle\ =\ A\,\sin\left(\,\int_{0}^{% t}\left(\omega_{c}+B\,\sin(\omega_{m}\,\tau)\right)d\tau\right)\\ &\displaystyle\ =\ A\,\sin\left(\omega_{c}\,t-\frac{B}{\omega_{m}}\left(\cos(% \omega_{m}\,t)-1\right)\right)\\ &\displaystyle\ =\ A\,\sin\left(\omega_{c}\,t+\frac{B}{\omega_{m}}\left(\sin(% \omega_{m}\,t-\pi/2)+1\right)\right)\\ \end{aligned}
  3. ϕ c = B / ω m \phi_{c}=B/\omega_{m}\,
  4. ϕ m = - π / 2 \phi_{m}=-\pi/2\,
  5. F M ( t ) \displaystyle FM(t)
  6. ω c , ω m \omega_{c}\,,\,\omega_{m}\,
  7. ω = 2 π f \,\omega=2\pi f\,
  8. β = B / ω m \beta=B/\omega_{m}\,
  9. J n ( β ) J_{n}(\beta)\,
  10. n n\,
  11. sin ( x ± y ) \displaystyle\sin(x\pm y)
  12. cos ( β sin θ ) \displaystyle\cos(\beta\sin\theta)
  13. sin ( θ c + β sin ( θ m ) ) \displaystyle\sin\left(\theta_{c}+\beta\,\sin(\theta_{m})\right)

Frequentist_probability.html

  1. n t n_{t}
  2. n x n_{x}
  3. x x
  4. P ( x ) P(x)
  5. P ( x ) n x n t . P(x)\approx\frac{n_{x}}{n_{t}}.
  6. P ( x ) = lim n t n x n t . P(x)=\lim_{n_{t}\rightarrow\infty}\frac{n_{x}}{n_{t}}.

Fresnel_equations.html

  1. θ i = θ r , \theta_{\mathrm{i}}=\theta_{\mathrm{r}},
  2. n 1 sin θ i = n 2 sin θ t . n_{1}\sin\theta_{\mathrm{i}}=n_{2}\sin\theta_{\mathrm{t}}.
  3. R s = | Z 2 cos θ i - Z 1 cos θ t Z 2 cos θ i + Z 1 cos θ t | 2 = | μ 2 ϵ 2 cos θ i - μ 1 ϵ 1 cos θ t μ 2 ϵ 2 cos θ i + μ 1 ϵ 1 cos θ t | 2 , R_{\mathrm{s}}=\left|\frac{Z_{2}\cos\theta_{\mathrm{i}}-Z_{1}\cos\theta_{% \mathrm{t}}}{Z_{2}\cos\theta_{\mathrm{i}}+Z_{1}\cos\theta_{\mathrm{t}}}\right|% ^{2}=\left|\frac{\sqrt{\frac{\mu_{2}}{\epsilon_{2}}}\cos\theta_{\mathrm{i}}-% \sqrt{\frac{\mu_{1}}{\epsilon_{1}}}\cos\theta_{\mathrm{t}}}{\sqrt{\frac{\mu_{2% }}{\epsilon_{2}}}\cos\theta_{\mathrm{i}}+\sqrt{\frac{\mu_{1}}{\epsilon_{1}}}% \cos\theta_{\mathrm{t}}}\right|^{2},
  4. R p = | Z 2 cos θ t - Z 1 cos θ i Z 2 cos θ t + Z 1 cos θ i | 2 = | μ 2 ϵ 2 cos θ t - μ 1 ϵ 1 cos θ i μ 2 ϵ 2 cos θ t + μ 1 ϵ 1 cos θ i | 2 , R_{\mathrm{p}}=\left|\frac{Z_{2}\cos\theta_{\mathrm{t}}-Z_{1}\cos\theta_{% \mathrm{i}}}{Z_{2}\cos\theta_{\mathrm{t}}+Z_{1}\cos\theta_{\mathrm{i}}}\right|% ^{2}=\left|\frac{\sqrt{\frac{\mu_{2}}{\epsilon_{2}}}\cos\theta_{\mathrm{t}}-% \sqrt{\frac{\mu_{1}}{\epsilon_{1}}}\cos\theta_{\mathrm{i}}}{\sqrt{\frac{\mu_{2% }}{\epsilon_{2}}}\cos\theta_{\mathrm{t}}+\sqrt{\frac{\mu_{1}}{\epsilon_{1}}}% \cos\theta_{\mathrm{i}}}\right|^{2},
  5. Z 1 = Z 0 n 1 , Z 2 = Z 0 n 2 . Z_{1}=\frac{Z_{0}}{n_{1}},\ Z_{2}=\frac{Z_{0}}{n_{2}}.
  6. R s = | n 1 cos θ i - n 2 cos θ t n 1 cos θ i + n 2 cos θ t | 2 = | n 1 cos θ i - n 2 1 - ( n 1 n 2 sin θ i ) 2 n 1 cos θ i + n 2 1 - ( n 1 n 2 sin θ i ) 2 | 2 , R_{\mathrm{s}}=\left|\frac{n_{1}\cos\theta_{\mathrm{i}}-n_{2}\cos\theta_{% \mathrm{t}}}{n_{1}\cos\theta_{\mathrm{i}}+n_{2}\cos\theta_{\mathrm{t}}}\right|% ^{2}=\left|\frac{n_{1}\cos\theta_{\mathrm{i}}-n_{2}\sqrt{1-\left(\frac{n_{1}}{% n_{2}}\sin\theta_{\mathrm{i}}\right)^{2}}}{n_{1}\cos\theta_{\mathrm{i}}+n_{2}% \sqrt{1-\left(\frac{n_{1}}{n_{2}}\sin\theta_{\mathrm{i}}\right)^{2}}}\right|^{% 2}\!,
  7. R p = | n 1 cos θ t - n 2 cos θ i n 1 cos θ t + n 2 cos θ i | 2 = | n 1 1 - ( n 1 n 2 sin θ i ) 2 - n 2 cos θ i n 1 1 - ( n 1 n 2 sin θ i ) 2 + n 2 cos θ i | 2 . R_{\mathrm{p}}=\left|\frac{n_{1}\cos\theta_{\mathrm{t}}-n_{2}\cos\theta_{% \mathrm{i}}}{n_{1}\cos\theta_{\mathrm{t}}+n_{2}\cos\theta_{\mathrm{i}}}\right|% ^{2}=\left|\frac{n_{1}\sqrt{1-\left(\frac{n_{1}}{n_{2}}\sin\theta_{\mathrm{i}}% \right)^{2}}-n_{2}\cos\theta_{\mathrm{i}}}{n_{1}\sqrt{1-\left(\frac{n_{1}}{n_{% 2}}\sin\theta_{\mathrm{i}}\right)^{2}}+n_{2}\cos\theta_{\mathrm{i}}}\right|^{2% }\!.
  8. T s = 1 - R s T_{\mathrm{s}}=1-R_{\mathrm{s}}
  9. T p = 1 - R p T_{\mathrm{p}}=1-R_{\mathrm{p}}
  10. R = R s + R p 2 . R=\frac{R_{\mathrm{s}}+R_{\mathrm{p}}}{2}.
  11. \bot
  12. \parallel
  13. r s = n 1 cos θ i - n 2 cos θ t n 1 cos θ i + n 2 cos θ t , r_{\mathrm{s}}=\frac{n_{1}\cos\theta_{\mathrm{i}}-n_{2}\cos\theta_{\mathrm{t}}% }{n_{1}\cos\theta_{\mathrm{i}}+n_{2}\cos\theta_{\mathrm{t}}},
  14. t s = 2 n 1 cos θ i n 1 cos θ i + n 2 cos θ t , t_{\mathrm{s}}=\frac{2n_{1}\cos\theta_{\mathrm{i}}}{n_{1}\cos\theta_{\mathrm{i% }}+n_{2}\cos\theta_{\mathrm{t}}},
  15. r p = n 2 cos θ i - n 1 cos θ t n 1 cos θ t + n 2 cos θ i , r_{\mathrm{p}}=\frac{n_{2}\cos\theta_{\mathrm{i}}-n_{1}\cos\theta_{\mathrm{t}}% }{n_{1}\cos\theta_{\mathrm{t}}+n_{2}\cos\theta_{\mathrm{i}}},
  16. t p = 2 n 1 cos θ i n 1 cos θ t + n 2 cos θ i . t_{\mathrm{p}}=\frac{2n_{1}\cos\theta_{\mathrm{i}}}{n_{1}\cos\theta_{\mathrm{t% }}+n_{2}\cos\theta_{\mathrm{i}}}.
  17. R = | r | 2 . R=|r|^{2}.
  18. T = n 2 cos θ t n 1 cos θ i | t | 2 . T=\frac{n_{2}\cos\theta_{\mathrm{t}}}{n_{1}\cos\theta_{\mathrm{i}}}|t|^{2}.
  19. ρ = n 2 n 1 , \rho=\frac{n_{2}}{n_{1}},
  20. T = ρ m | t | 2 . T=\rho m|t|^{2}.

Fresnel_zone.html

  1. F n = n λ d 1 d 2 d 1 + d 2 F_{n}=\sqrt{\frac{n\lambda d_{1}d_{2}}{d_{1}+d_{2}}}
  2. λ \lambda
  3. d 1 = d 2 d_{1}=d_{2}
  4. D = d 1 + d 2 D=d_{1}+d_{2}
  5. λ = c f \lambda=\frac{c}{f}
  6. r = 8.657 D f r=8.657\sqrt{{D}\over f}
  7. r = 36.03 D f r=36.03\sqrt{{D}\over f}

Friction.html

  1. F f μ F n F_{\mathrm{f}}\leq\mu F_{\mathrm{n}}
  2. F f F_{\mathrm{f}}\,
  3. μ \mu\,
  4. F n F_{\mathrm{n}}\,
  5. F f F_{\mathrm{f}}\,
  6. μ F n \mu F_{\mathrm{n}}\,
  7. N = m g N=mg\,
  8. N N\,
  9. μ = μ s \mu=\mu_{\mathrm{s}}\,
  10. μ s \mu_{\mathrm{s}}\,
  11. μ = μ k \mu=\mu_{\mathrm{k}}\,
  12. μ k \mu_{\mathrm{k}}\,
  13. F f F_{\mathrm{f}}\,
  14. μ s \mu_{\mathrm{s}}\,
  15. μ k \mu_{\mathrm{k}}\,
  16. F m a x = μ s F n F_{max}=\mu_{\mathrm{s}}F_{n}\,
  17. F m a x F_{max}\,
  18. F m a x F_{max}\,
  19. F m a x F_{max}\,
  20. F k = μ k F n F_{k}=\mu_{\mathrm{k}}F_{n}\,
  21. tan θ = μ s \tan{\theta}=\mu_{\mathrm{s}}\,
  22. E t h = C 𝐅 fric ( 𝐱 ) d 𝐱 = C μ k 𝐅 n ( 𝐱 ) d 𝐱 E_{th}=\int_{C}\mathbf{F}_{\mathrm{fric}}(\mathbf{x})\cdot d\mathbf{x}\ =\int_% {C}\mu_{\mathrm{k}}\ \mathbf{F}_{\mathrm{n}}(\mathbf{x})\cdot d\mathbf{x}\,
  23. 𝐅 fric \mathbf{F}_{\mathrm{fric}}\,
  24. 𝐅 n \mathbf{F}_{\mathrm{n}}\,
  25. μ k \mu_{\mathrm{k}}\,
  26. 𝐱 \mathbf{x}\,

Full_moon.html

  1. d = 20.362955 + 29.530588861 × N + 102.026 × 10 - 12 × N 2 d=20.362955+29.530588861\times N+102.026\times 10^{-12}\times N^{2}
  2. - 0.000739 - ( 235 × 10 - 12 ) × N 2 -0.000739-(235\times 10^{-12})\times N^{2}

Full_width_at_half_maximum.html

  1. f ( x ) = 1 σ 2 π exp [ - ( x - x 0 ) 2 2 σ 2 ] f(x)=\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{(x-x_{0})^{2}}{2\sigma^{2}}\right]
  2. σ \sigma
  3. x 0 x_{0}
  4. FWHM = 2 2 ln 2 σ 2.355 σ . \mathrm{FWHM}=2\sqrt{2\ln 2}\;\sigma\approx 2.355\;\sigma.
  5. f ( x ) = 1 π γ [ 1 + ( x - x 0 γ ) 2 ] f(x)=\frac{1}{\pi\gamma\left[1+\left(\frac{x-x_{0}}{\gamma}\right)^{2}\right]}
  6. FWHM = 2 γ \mathrm{FWHM}=2\gamma
  7. f ( x ) = sech ( x X ) . f(x)=\operatorname{sech}\left(\frac{x}{X}\right).
  8. FWHM = 2 arsech ( 1 2 ) X = 2 ln ( 2 + 3 ) X 2.634 X \mathrm{FWHM}=2\;\operatorname{arsech}\left(\frac{1}{2}\right)X=2\ln(2+\sqrt{3% })\;X\approx 2.634\;X

Functional_analysis.html

  1. 2 ( 0 ) \ell^{\,2}(\aleph_{0})\,
  2. L p L^{\,p}
  3. p 1 p\geq 1
  4. μ \mu
  5. X X
  6. L p ( X ) L^{\,p}(X)
  7. L p ( X , μ ) L^{\,p}(X,\mu)
  8. L p ( μ ) L^{\,p}(\mu)
  9. [ f ] [\,f\,]
  10. p p
  11. f f\,
  12. X | f ( x ) | p d μ ( x ) < + \int_{X}\left|f(x)\right|^{p}\,d\mu(x)<+\infty
  13. μ \mu
  14. x X | f ( x ) | p < + \sum_{x\in X}\left|f(x)\right|^{p}<+\infty
  15. p ( X ) \ell^{\ p}(X)
  16. p \,\ell^{\,p\ }
  17. X X
  18. sup T F T ( x ) Y < , \sup\nolimits_{T\in F}\|T(x)\|_{Y}<\infty,
  19. sup T F T B ( X , Y ) < . \sup\nolimits_{T\in F}\|T\|_{B(X,Y)}<\infty.
  20. [ A φ ] ( t ) = t φ ( t ) . [A\varphi](t)=t\varphi(t).\;
  21. U * T U = A U^{*}TU=A\;
  22. [ T φ ] ( x ) = f ( x ) φ ( x ) . [T\varphi](x)=f(x)\varphi(x).\;
  23. T = f \|T\|=\|f\|_{\infty}
  24. f f
  25. p : V 𝐑 p:V→\mathbf{R}
  26. φ : U 𝐑 φ:U→\mathbf{R}
  27. U V U⊆V
  28. p p
  29. U U
  30. φ ( x ) p ( x ) x U \varphi(x)\leq p(x)\qquad\forall x\in U
  31. ψ : V 𝐑 ψ:V→\mathbf{R}
  32. φ φ
  33. V V
  34. ψ ψ
  35. ψ ( x ) = φ ( x ) x U , \psi(x)=\varphi(x)\qquad\forall x\in U,
  36. ψ ( x ) p ( x ) x V . \psi(x)\leq p(x)\qquad\forall x\in V.

Functional_decomposition.html

  1. y = f ( x 1 , x 2 , , x n ) y=f(x_{1},x_{2},\dots,x_{n})
  2. { g 1 , g 2 , g m } \{g_{1},g_{2},\dots g_{m}\}
  3. f ( x 1 , x 2 , , x n ) = ϕ ( g 1 ( x 1 , x 2 , , x n ) , g 2 ( x 1 , x 2 , , x n ) , g m ( x 1 , x 2 , , x n ) ) f(x_{1},x_{2},\dots,x_{n})=\phi(g_{1}(x_{1},x_{2},\dots,x_{n}),g_{2}(x_{1},x_{% 2},\dots,x_{n}),\dots g_{m}(x_{1},x_{2},\dots,x_{n}))
  4. ϕ \phi
  5. f f
  6. { g 1 , g 2 , g m } \{g_{1},g_{2},\dots g_{m}\}
  7. g i g_{i}
  8. { h 1 , h 2 , h p } \{h_{1},h_{2},\dots h_{p}\}
  9. g i ( x 1 , x 2 , , x n ) = γ ( h 1 ( x 1 , x 2 , , x n ) , h 2 ( x 1 , x 2 , , x n ) , h p ( x 1 , x 2 , , x n ) ) g_{i}(x_{1},x_{2},\dots,x_{n})=\gamma(h_{1}(x_{1},x_{2},\dots,x_{n}),h_{2}(x_{% 1},x_{2},\dots,x_{n}),\dots h_{p}(x_{1},x_{2},\dots,x_{n}))
  10. γ \gamma
  11. x 1 = f ( x 2 , x 3 , , x 6 ) x_{1}=f(x_{2},x_{3},\dots,x_{6})
  12. { g 1 , g 2 , g 3 } \{g_{1},g_{2},g_{3}\}
  13. x 1 = g 1 ( x 2 ) x_{1}=g_{1}(x_{2})
  14. x 2 = g 2 ( x 3 , x 4 , x 5 ) x_{2}=g_{2}(x_{3},x_{4},x_{5})
  15. x 5 = g 3 ( x 6 ) x_{5}=g_{3}(x_{6})
  16. a + b a+b
  17. a × b , a\times b,
  18. - a -a
  19. 1 / a . 1/a.
  20. a - b = a + ( - b ) , a-b=a+(-b),
  21. a ÷ b = a × ( 1 / b ) . a\div b=a\times(1/b).
  22. x 1 = f ( x 2 , x 3 , , x 6 ) x_{1}=f(x_{2},x_{3},\dots,x_{6})
  23. 4 5 = 1024 4^{5}=1024
  24. { x 2 , x 3 , , x 6 } \{x_{2},x_{3},\dots,x_{6}\}
  25. { x 2 , x 3 , , x 6 } \{x_{2},x_{3},\dots,x_{6}\}
  26. { g 1 , g 2 , g 3 } \{g_{1},g_{2},g_{3}\}
  27. g 1 = g 1 ( x 2 ) g_{1}=g_{1}(x_{2})
  28. g 2 = g 2 ( x 3 , x 4 , x 5 ) g_{2}=g_{2}(x_{3},x_{4},x_{5})
  29. 4 3 = 64 4^{3}=64
  30. g 3 = g 3 ( x 6 ) g_{3}=g_{3}(x_{6})
  31. 4 + 64 + 4 = 72 4+64+4=72
  32. x 1 x_{1}
  33. x 2 x_{2}
  34. x 2 x_{2}
  35. x 1 x_{1}
  36. x 1 {x_{1}}
  37. x 1 {x_{1}}
  38. x 1 {x_{1}}
  39. x 2 {x_{2}}
  40. x 1 {x_{1}}
  41. x 2 {x_{2}}
  42. x 1 {x_{1}}
  43. x 2 {x_{2}}
  44. { x 3 , x 4 , x 5 } \{x_{3},x_{4},x_{5}\}
  45. x 2 {x_{2}}
  46. e state of the Emacs editor and running operating system e\,\equiv\,\text{state of the Emacs editor and running operating system}
  47. e e with some component/part of its state changed e^{\prime}\,\equiv e\,\text{ with some component/part of its state changed}
  48. f : ( e , l i s p e x p r e s s i o n ) e f:(e,lisp\,\,expression)\rightarrow e^{\prime}
  49. f r o m E x p r : l i s p e x p r e s s i o n { o b j e c t , if success e r r o r , if failure fromExpr:lisp\,\,expression\rightarrow\begin{cases}object,&\,\text{if success}% \\ error,&\,\text{if failure}\end{cases}
  50. e v a l u a t e : ( o b j e c t , e ) e evaluate:(object,e)\rightarrow e^{\prime}
  51. p r i n t : ( e r r o r , e ) e print:(error,e)\rightarrow e^{\prime}
  52. f ( t ) f(t)
  53. f ( t ) f(t)
  54. f ( t ) = a 1 g 1 ( t ) + a 2 g 2 ( t ) + a 3 g 3 ( t ) + + a n g n ( t ) f(t)=a_{1}\cdot g_{1}(t)+a_{2}\cdot g_{2}(t)+a_{3}\cdot g_{3}(t)+\dots+a_{n}% \cdot g_{n}(t)
  55. { g 1 ( t ) , g 2 ( t ) , g 3 ( t ) , , g n ( t ) } \{g_{1}(t),g_{2}(t),g_{3}(t),\dots,g_{n}(t)\}
  56. { a 1 , a 2 , a 3 , , a n } \{a_{1},a_{2},a_{3},\dots,a_{n}\}
  57. T { } T\{\}
  58. T { f ( t ) } T\{f(t)\}
  59. T { f ( t ) } = T { a 1 g 1 ( t ) + a 2 g 2 ( t ) + a 3 g 3 ( t ) + + a n g n ( t ) } T\{f(t)\}=T\{a_{1}\cdot g_{1}(t)+a_{2}\cdot g_{2}(t)+a_{3}\cdot g_{3}(t)+\dots% +a_{n}\cdot g_{n}(t)\}
  60. = a 1 T { g 1 ( t ) } + a 2 T { g 2 ( t ) } + a 3 T { g 3 ( t ) } + + a n T { g n ( t ) } =a_{1}\cdot T\{g_{1}(t)\}+a_{2}\cdot T\{g_{2}(t)\}+a_{3}\cdot T\{g_{3}(t)\}+% \dots+a_{n}\cdot T\{g_{n}(t)\}

Functional_programming.html

  1. d / d x d/dx
  2. f f

Functor.html

  1. X C X\in C
  2. F ( X ) D F(X)\in D
  3. f : X Y C f:X\rightarrow Y\in C
  4. F ( f ) : F ( X ) F ( Y ) D F(f):F(X)\rightarrow F(Y)\in D
  5. F ( id X ) = id F ( X ) F(\mathrm{id}_{X})=\mathrm{id}_{F(X)}\,\!
  6. X C X\in C
  7. F ( g f ) = F ( g ) F ( f ) F(g\circ f)=F(g)\circ F(f)
  8. f : X Y f:X\rightarrow Y\,\!
  9. g : Y Z . g:Y\rightarrow Z.\,\!
  10. X C X\in C
  11. F ( X ) D , F(X)\in D,
  12. f : X Y C f:X\rightarrow Y\in C
  13. F ( f ) : F ( Y ) F ( X ) D F(f):F(Y)\rightarrow F(X)\in D
  14. F ( id X ) = id F ( X ) F(\mathrm{id}_{X})=\mathrm{id}_{F(X)}\,\!
  15. X C X\in C
  16. F ( g f ) = F ( f ) F ( g ) F(g\circ f)=F(f)\circ F(g)
  17. f : X Y f:X\rightarrow Y\,\!
  18. g : Y Z . g:Y\rightarrow Z.\,\!
  19. C op C^{\mathrm{op}}
  20. F : C D F:C\rightarrow D
  21. F : C op D F:C^{\mathrm{op}}\rightarrow D
  22. F : C D op F:C\rightarrow D^{\mathrm{op}}
  23. F : C D F:C\rightarrow D
  24. F op : C op D op F^{\mathrm{op}}:C^{\mathrm{op}}\rightarrow D^{\mathrm{op}}
  25. C op C^{\mathrm{op}}
  26. D op D^{\mathrm{op}}
  27. C C
  28. D D
  29. F op F^{\mathrm{op}}
  30. F F
  31. C op C^{\mathrm{op}}
  32. C C
  33. D D
  34. F op F^{\mathrm{op}}
  35. F F
  36. F : C 0 C 1 F:C_{0}\rightarrow C_{1}
  37. G : C 1 op C 2 G:C_{1}^{\mathrm{op}}\rightarrow C_{2}
  38. G F op G\circ F^{\mathrm{op}}
  39. G op F G^{\mathrm{op}}\circ F
  40. ( F op ) op = F (F^{\mathrm{op}})^{\mathrm{op}}=F
  41. D : J C D:J\rightarrow C
  42. D : C J D:C\rightarrow J
  43. U V U\subseteq V
  44. f : X Y f:X\to Y
  45. U X U\subseteq X
  46. f ( U ) Y f(U)\subseteq Y
  47. f : X Y f:X\to Y
  48. V Y V\subseteq Y
  49. f - 1 ( V ) X . f^{-1}(V)\subseteq X.
  50. V W V\otimes W
  51. \mapsto
  52. \mapsto

Fundamental_frequency.html

  1. x ( t ) = x ( t + T ) for all t x(t)=x(t+T)\,\text{ for all }t\in\mathbb{R}
  2. f 0 = 1 T f_{0}=\frac{1}{T}
  3. λ 0 = 4 L . \lambda_{0}=4L.
  4. λ 0 = v f 0 \lambda_{0}=\frac{v}{f_{0}}
  5. f 0 = v 4 L . f_{0}=\frac{v}{4L}.
  6. f 0 = v 2 L . f_{0}=\frac{v}{2L}.
  7. ω n 2 = k m \omega_{\mathrm{n}}^{2}=\frac{k}{m}\,
  8. f n = 1 2 π k m f_{\mathrm{n}}=\frac{1}{2\pi}\sqrt{\frac{k}{m}}\,

Fundamental_group.html

  1. { f : [ 0 , 1 ] X : f ( 0 ) = x 0 = f ( 1 ) } \{f:[0,1]\to X:\ f(0)=x_{0}=f(1)\}
  2. { f : [ 0 , 1 ] X : f ( 0 ) = x 0 = f ( 1 ) } / h \{f:[0,1]\to X:\ f(0)=x_{0}=f(1)\}/h
  3. ( f * g ) ( t ) = { f ( 2 t ) 0 t 1 2 g ( 2 t - 1 ) 1 2 t 1 (f*g)(t)=\begin{cases}f(2t)&0\leq t\leq\tfrac{1}{2}\\ g(2t-1)&\tfrac{1}{2}\leq t\leq 1\end{cases}
  4. π 1 ( X , x 0 ) , \pi_{1}(X,x_{0}),
  5. B 3 B_{3}
  6. f * : π 1 ( X , x 0 ) π 1 ( Y , y 0 ) . f_{*}:\pi_{1}(X,x_{0})\to\pi_{1}(Y,y_{0}).
  7. X Y π 1 ( X , x 0 ) π 1 ( Y , y 0 ) . X\simeq Y\Rightarrow\pi_{1}(X,x_{0})\cong\pi_{1}(Y,y_{0}).
  8. π 1 ( X × Y ) π 1 ( X ) × π 1 ( Y ) \pi_{1}(X\times Y)\cong\pi_{1}(X)\times\pi_{1}(Y)
  9. π 1 ( X Y ) π 1 ( X ) * π 1 ( Y ) . \pi_{1}(X\vee Y)\cong\pi_{1}(X)*\pi_{1}(Y).
  10. \vee
  11. F E B . F\to E\to B.
  12. 1 Γ G H 1. 1\to\Gamma\to G\to H\to 1.
  13. X X
  14. f : [ 0 , r ] X f:[0,r]\to X
  15. r 0 r\geqslant 0
  16. f ( 0 ) , f ( r ) f(0),f(r)
  17. r r
  18. f f
  19. g : [ 0 , s ] X g:[0,s]\to X
  20. f ( r ) = g ( 0 ) f(r)=g(0)
  21. f g : [ 0 , r + s ] X f\circ g:[0,r+s]\to X
  22. f f
  23. [ 0 , r ] [0,r]
  24. [ r , r + s ] [r,r+s]
  25. X X
  26. f : [ 0 , r ] X , g : [ 0 , s ] X f:[0,r]\to X,g:[0,s]\to X
  27. u , v 0 u,v\geqslant 0
  28. r + u = s + v r+u=s+v
  29. f u , g v : [ 0 , r + u ] X f\circ u,g\circ v:[0,r+u]\to X
  30. f u f\circ u
  31. f f
  32. u u

Fundamental_interaction.html

  1. ħ / 2 {ħ}/{2}
  2. 1 {1}
  3. 1 r 2 \frac{1}{r^{2}}
  4. 1 r e - m W , Z r \frac{1}{r}\ e^{-m_{W,Z}\ r}
  5. 1 r 2 \frac{1}{r^{2}}
  6. 4000 g H 2 O 1 mol H 2 O 18 g H 2 O 10 mol e - 1 mol H 2 O 96 , 000 C 1 mol e - = 2.1 × 10 8 C 4000\ \mbox{g}~{}\,H_{2}O\cdot\frac{1\ \mbox{mol}~{}\,H_{2}O}{18\ \mbox{g}~{}% \,H_{2}O}\cdot\frac{10\ \mbox{mol}~{}\,e^{-}}{1\ \mbox{mol}~{}\,H_{2}O}\cdot% \frac{96,000\ \mbox{C}~{}\,}{1\ \mbox{mol}~{}\,e^{-}}=2.1\times 10^{8}C\ \,
  7. 1 4 π ε 0 ( 2.1 × 10 8 C ) 2 ( 1 m ) 2 = 4.1 × 10 26 N . {1\over 4\pi\varepsilon_{0}}\frac{(2.1\times 10^{8}C)^{2}}{(1m)^{2}}=4.1\times 1% 0^{26}N.

Fundamental_theorem_of_algebra.html

  1. ( x 2 - ( 2 + α ) x + 1 + 7 + α ) ( x 2 - ( 2 - α ) x + 1 + 7 - α ) , (x^{2}-(2+\alpha)x+1+\sqrt{7}+\alpha)(x^{2}-(2-\alpha)x+1+\sqrt{7}-\alpha),
  2. x 4 + a 4 = ( x 2 + a 2 x + a 2 ) ( x 2 - a 2 x + a 2 ) . x^{4}+a^{4}=(x^{2}+a\sqrt{2}\cdot x+a^{2})(x^{2}-a\sqrt{2}\cdot x+a^{2}).\,
  3. q ( z ) = p ( z ) p ( z ¯ ) ¯ q(z)=p(z)\overline{p(\overline{z})}
  4. 1 2 | z n | < | p ( z ) | < 3 2 | z n | \tfrac{1}{2}|z^{n}|<|p(z)|<\tfrac{3}{2}|z^{n}|
  5. p ( z ) = a + c k ( z - z 0 ) k + c k + 1 ( z - z 0 ) k + 1 + + c n ( z - z 0 ) n . p(z)=a+c_{k}(z-z_{0})^{k}+c_{k+1}(z-z_{0})^{k+1}+\ldots+c_{n}(z-z_{0})^{n}.
  6. q ( z ) = a + c k ( z - z 0 ) k q(z)=a+c_{k}(z-z_{0})^{k}
  7. | p ( z ) - q ( z ) ( z - z 0 ) k + 1 | \left|\frac{p(z)-q(z)}{(z-z_{0})^{k+1}}\right|
  8. θ 0 = ( arg ( a ) + π - arg ( c k ) ) / k \theta_{0}=(\arg(a)+\pi-\arg(c_{k}))/k
  9. z = z 0 + r e i θ 0 z=z_{0}+re^{i\theta_{0}}
  10. | p ( z ) | < | q ( z ) | + r k + 1 | p ( z ) - q ( z ) r k + 1 | | a + ( - 1 ) c k r k e i ( arg ( a ) - arg ( c k ) ) | + M r k + 1 = | a | - | c k | r k + M r k + 1 . \begin{aligned}\displaystyle|p(z)|&\displaystyle<|q(z)|+r^{k+1}\left|\frac{p(z% )-q(z)}{r^{k+1}}\right|\\ &\displaystyle\leq\left|a+(-1)c_{k}r^{k}e^{i(\arg(a)-\arg(c_{k}))}\right|+Mr^{% k+1}\\ &\displaystyle=|a|-|c_{k}|r^{k}+Mr^{k+1}.\end{aligned}
  11. 1 2 π i c ( r ) p ( z ) p ( z ) d z , \frac{1}{2\pi i}\int_{c(r)}\frac{p^{\prime}(z)}{p(z)}\,dz,
  12. 1 2 π i c ( r ) ( p ( z ) p ( z ) - n z ) d z = 1 2 π i c ( r ) z p ( z ) - n p ( z ) z p ( z ) d z . \frac{1}{2\pi i}\int_{c(r)}\left(\frac{p^{\prime}(z)}{p(z)}-\frac{n}{z}\right)% dz=\frac{1}{2\pi i}\int_{c(r)}\frac{zp^{\prime}(z)-np(z)}{zp(z)}\,dz.
  13. R ( z ) = ( z I n - A ) - 1 , R(z)=(zI_{n}-A)^{-1},
  14. c ( r ) R ( z ) d z = 0. \int_{c(r)}R(z)dz=0.
  15. R ( z ) = z - 1 ( I n - z - 1 A ) - 1 = z - 1 k = 0 1 z k A k R(z)=z^{-1}(I_{n}-z^{-1}A)^{-1}=z^{-1}\sum_{k=0}^{\infty}\frac{1}{z^{k}}A^{k}\cdot
  16. c ( r ) R ( z ) d z = k = 0 c ( r ) d z z k + 1 A k = 2 π i I n \int_{c(r)}R(z)dz=\sum_{k=0}^{\infty}\int_{c(r)}\frac{dz}{z^{k+1}}A^{k}=2\pi iI% _{n}
  17. p ( z ) = p ( z 0 ) + c k ( z - z 0 ) k + c k + 1 ( z - z 0 ) k + 1 + + c n ( z - z 0 ) n . p(z)=p(z_{0})+c_{k}(z-z_{0})^{k}+c_{k+1}(z-z_{0})^{k+1}+\cdots+c_{n}(z-z_{0})^% {n}.
  18. p ( z ) = a ( z - z 1 ) ( z - z 2 ) ( z - z n ) . p(z)=a(z-z_{1})(z-z_{2})\cdots(z-z_{n}).
  19. q t ( z ) = 1 i < j n ( z - z i - z j - t z i z j ) . q_{t}(z)=\prod_{1\leq i<j\leq n}\left(z-z_{i}-z_{j}-tz_{i}z_{j}\right).\,
  20. 𝐒 2 K g = 4 π \int_{\mathbf{S}^{2}}K_{g}=4\pi
  21. f ( 1 w ) = p ( 1 w ) p * ( 1 w ) = w - 2 n p * ( w ) p ( w ) = w - 2 n f ( w ) f(\tfrac{1}{w})=p(\tfrac{1}{w})p^{*}(\tfrac{1}{w})=w^{-2n}p^{*}(w)p(w)=w^{-2n}% f(w)
  22. g = 1 | f ( w ) | 2 n | d w | 2 g=\frac{1}{|f(w)|^{\frac{2}{n}}}\,|dw|^{2}
  23. g = 1 | f ( 1 / w ) | 2 n | d ( 1 / w ) | 2 g=\frac{1}{|f(1/w)|^{\frac{2}{n}}}\,|d(1/w)|^{2}
  24. w 𝐂 : 1 | f ( w ) | 1 n K g = 1 n Δ log | f ( w ) | = 1 n Δ Re ( log f ( w ) ) = 0 \forall w\in\mathbf{C}:\frac{1}{|f(w)|^{\frac{1}{n}}}\,K_{g}=\frac{1}{n}\Delta% \log|f(w)|=\frac{1}{n}\Delta\,\text{Re}(\log f(w))=0
  25. z n + a n - 1 z n - 1 + + a 1 z + a 0 \scriptstyle z^{n}+a_{n-1}z^{n-1}+\cdots+a_{1}z+a_{0}
  26. R := 1 + max { | a 0 | , , | a n - 1 | } . R_{\infty}:=1+\max\{|a_{0}|,\cdots,|a_{n-1}|\}.
  27. a := ( a 0 , a 1 , , a n - 1 ) \scriptstyle a:=(\ a_{0},\ a_{1},...,\ a_{n-1})
  28. ( 1 , a p ) \scriptstyle(1,\|a\|_{p})
  29. R 1 := max { 1 , 0 k < n | a k | } , R_{1}:=\max\Bigl\{1,\sum_{0\leq k<n}|a_{k}|\Bigr\},
  30. R p := [ 1 + ( 0 k < n | a k | p ) q p ] 1 q , R_{p}:=\biggl[1+\Bigl(\sum_{0\leq k<n}|a_{k}|^{p}\Bigr)^{\frac{q}{p}}\biggr]^{% \frac{1}{q}},
  31. P ( z ) := a n z n + a n - 1 z n - 1 + + a 1 z + a 0 \scriptstyle P(z):=a_{n}z^{n}+a_{n-1}z^{n-1}+\cdots+a_{1}z+a_{0}
  32. 1 ζ \scriptstyle\frac{1}{\zeta}
  33. a 0 z n + a 1 z n - 1 + + a n - 1 z + a n \scriptstyle a_{0}z^{n}+a_{1}z^{n-1}+\cdots+a_{n-1}z+a_{n}
  34. | ζ - ζ 0 | |\zeta-\zeta_{0}|
  35. ζ 0 \zeta_{0}
  36. ζ - ζ 0 \zeta-\zeta_{0}
  37. P ( z + ζ 0 ) P(z+\zeta_{0})
  38. z = ζ 0 . z=\zeta_{0}.
  39. z n + a n - 1 z n - 1 + + a 1 z + a 0 \scriptstyle z^{n}+a_{n-1}z^{n-1}+\cdots+a_{1}z+a_{0}
  40. - ζ n = a n - 1 ζ n - 1 + + a 1 ζ + a 0 \scriptstyle-\zeta^{n}=a_{n-1}\zeta^{n-1}+\cdots+a_{1}\zeta+a_{0}
  41. | ζ | n a p ( ζ n - 1 , , ζ , 1 ) q \scriptstyle|\zeta|^{n}\leq\|a\|_{p}\|(\zeta^{n-1},\cdots,\zeta,1)\|_{q}
  42. | ζ | n a 1 max { | ζ | n - 1 , , | ζ | , 1 } = a 1 | ζ | n - 1 \scriptstyle|\zeta|^{n}\leq\|a\|_{1}\max\{|\zeta|^{n-1},\cdots,|\zeta|,1\}=\|a% \|_{1}|\zeta|^{n-1}
  43. | ζ | max { 1 , a 1 } \scriptstyle|\zeta|\leq\max\{1,\|a\|_{1}\}
  44. | ζ | n q a p q | ζ | q n | ζ | q - 1 \scriptstyle|\zeta|^{nq}\leq\|a\|_{p}^{q}\frac{|\zeta|^{qn}}{|\zeta|^{q}-1}
  45. | ζ | q 1 + a p q \scriptstyle|\zeta|^{q}\leq 1+\|a\|_{p}^{q}
  46. | ζ | ( 1 , a p ) q = R p \scriptstyle|\zeta|\leq\|(1,\|a\|_{p})\|_{q}=R_{p}

Fundamental_theorem_of_arithmetic.html

  1. 4 {}^{4}
  2. 1 {}^{1}
  3. 2 {}^{2}
  4. n = p 1 α 1 p 2 α 2 p k α k = i = 1 k p i α i n=p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}\cdots p_{k}^{\alpha_{k}}=\prod_{i=1}^{k% }p_{i}^{\alpha_{i}}
  5. n = 2 n 2 3 n 3 5 n 5 7 n 7 = p i n p i , n=2^{n_{2}}3^{n_{3}}5^{n_{5}}7^{n_{7}}\cdots=\prod p_{i}^{n_{p_{i}}},
  6. a b = 2 a 2 + b 2 3 a 3 + b 3 5 a 5 + b 5 7 a 7 + b 7 = p i a p i + b p i , a\cdot b=2^{a_{2}+b_{2}}\,3^{a_{3}+b_{3}}\,5^{a_{5}+b_{5}}\,7^{a_{7}+b_{7}}% \cdots=\prod p_{i}^{a_{p_{i}}+b_{p_{i}}},
  7. gcd ( a , b ) = 2 min ( a 2 , b 2 ) 3 min ( a 3 , b 3 ) 5 min ( a 5 , b 5 ) 7 min ( a 7 , b 7 ) = p i min ( a p i , b p i ) , \gcd(a,b)=2^{\min(a_{2},b_{2})}\,3^{\min(a_{3},b_{3})}\,5^{\min(a_{5},b_{5})}% \,7^{\min(a_{7},b_{7})}\cdots=\prod p_{i}^{\min(a_{p_{i}},b_{p_{i}})},
  8. lcm ( a , b ) = 2 max ( a 2 , b 2 ) 3 max ( a 3 , b 3 ) 5 max ( a 5 , b 5 ) 7 max ( a 7 , b 7 ) = p i max ( a p i , b p i ) . \operatorname{lcm}(a,b)=2^{\max(a_{2},b_{2})}\,3^{\max(a_{3},b_{3})}\,5^{\max(% a_{5},b_{5})}\,7^{\max(a_{7},b_{7})}\cdots=\prod p_{i}^{\max(a_{p_{i}},b_{p_{i% }})}.
  9. s \displaystyle s
  10. s p 1 = p 2 p m = q 2 q n . \begin{aligned}\displaystyle\frac{s}{p_{1}}&\displaystyle=p_{2}\cdots p_{m}\\ &\displaystyle=q_{2}\cdots q_{n}.\end{aligned}
  11. s p 1 p 2 = p 3 p m = q 3 q n . \begin{aligned}\displaystyle\frac{s}{p_{1}p_{2}}&\displaystyle=p_{3}\cdots p_{% m}\\ &\displaystyle=q_{3}\cdots q_{n}.\end{aligned}
  12. p p
  13. q q
  14. s \displaystyle s
  15. t = ( q 1 - p 1 ) ( q 2 q n ) , t=(q_{1}-p_{1})(q_{2}\cdots q_{n}),
  16. t = ( r 1 r h ) ( q 2 q n ) , t=(r_{1}\cdots r_{h})(q_{2}\cdots q_{n}),
  17. p 1 ( k + 1 ) = q 1 . p_{1}(k+1)=q_{1}.
  18. [ i ] . \mathbb{Z}[i].
  19. [ ω ] \mathbb{Z}[\omega]
  20. ω = - 1 + - 3 2 , \omega=\frac{-1+\sqrt{-3}}{2},
  21. ω 3 = 1 \omega^{3}=1
  22. ± 1 , ± ω , ± ω 2 \pm 1,\pm\omega,\pm\omega^{2}
  23. [ - 5 ] \mathbb{Z}[\sqrt{-5}]
  24. 6 = 2 × 3 = ( 1 + - 5 ) × ( 1 - - 5 ) . 6=2\times 3=(1+\sqrt{-5})\times(1-\sqrt{-5}).
  25. [ - 5 ] \mathbb{Z}[\sqrt{-5}]
  26. [ - 5 ] \mathbb{Z}[\sqrt{-5}]
  27. [ - 5 ] \mathbb{Z}[\sqrt{-5}]
  28. [ - 5 ] \mathbb{Z}[\sqrt{-5}]
  29. . \mathbb{Z}.
  30. \mathbb{Z}
  31. [ - 5 ] . \mathbb{Z}[\sqrt{-5}].
  32. \mathbb{Z}
  33. [ i ] \mathbb{Z}[i]
  34. [ ω ] , \mathbb{Z}[\omega],
  35. [ - 5 ] . \mathbb{Z}[\sqrt{-5}].

Fusion_power.html

  1. E = m c 2 E=mc^{2}
  2. P fusion = n A n B σ v A , B E fusion P\text{fusion}=n_{A}n_{B}\langle\sigma v_{A,B}\rangle E\text{fusion}
  3. P fusion P\text{fusion}
  4. σ v A , B \langle\sigma v_{A,B}\rangle
  5. E fusion E\text{fusion}
  6. ( n t τ ) (nt\tau)
  7. β = p p m a g = n k B T ( B 2 / 2 μ 0 ) \beta=\frac{p}{p_{mag}}=\frac{nk_{B}T}{(B^{2}/2\mu_{0})}

Fusor.html

  1. P fusion = n A n B σ v A , B E fusion P\text{fusion}=n_{A}n_{B}\langle\sigma v_{A,B}\rangle E\text{fusion}
  2. P fusion P\text{fusion}
  3. σ v A , B \langle\sigma v_{A,B}\rangle
  4. E fusion E\text{fusion}
  5. P out = η capture ( P fusion - P conduction - P radiation ) P\text{out}=\eta\text{capture}\left(P\text{fusion}-P\text{conduction}-P\text{% radiation}\right)

Fuzzy_control_system.html

  1. m u ( 1 ) . o u t p u t ( 1 ) + m u ( 2 ) . o u t p u t ( 2 ) + m u ( 3 ) . o u t p u t ( 3 ) + m u ( 4 ) . o u t p u t ( 4 ) m u ( 1 ) + m u ( 2 ) + m u ( 3 ) + m u ( 4 ) \frac{mu(1).output(1)+mu(2).output(2)+mu(3).output(3)+mu(4).output(4)}{mu(1)+% mu(2)+mu(3)+mu(4)}
  2. = ( 0.3 * 0 ) + ( 0.7 * - 0.5 ) + ( 0 * 1 ) + ( 0.3 * - 1 ) 0.3 + 0.7 + 0 + 0.3 =\frac{(0.3*0)+(0.7*-0.5)+(0*1)+(0.3*-1)}{0.3+0.7+0+0.3}
  3. = - 0.5 =-0.5
  4. X 0 X_{0}
  5. X i X_{i}
  6. M i M_{i}
  7. 0 = ( X 1 - X 0 ) * M 1 + ( X 2 - X 0 ) * M 2 + + ( X n - X 0 ) * M n 0=(X_{1}-X_{0})*M_{1}+(X_{2}-X_{0})*M_{2}+\ldots+(X_{n}-X_{0})*M_{n}
  8. 0 = ( X 1 * M 1 + X 2 * M 2 + + X n * M n ) - X 0 * ( M 1 + M 2 + + M n ) 0=(X_{1}*M_{1}+X_{2}*M_{2}+\ldots+X_{n}*M_{n})-X_{0}*(M_{1}+M_{2}+\ldots+M_{n})
  9. X 0 * ( M 1 + M 2 + + M n ) = X 1 * M 1 + X 2 * M 2 + + X n * M n X_{0}*(M_{1}+M_{2}+\ldots+M_{n})=X_{1}*M_{1}+X_{2}*M_{2}+\ldots+X_{n}*M_{n}
  10. X 0 = X 1 * M 1 + X 2 * M 2 + + X n * M n M 1 + M 2 + + M n X_{0}=\frac{X_{1}*M_{1}+X_{2}*M_{2}+\ldots+X_{n}*M_{n}}{M_{1}+M_{2}+\ldots+M_{% n}}

Fuzzy_logic.html

  1. \rightarrow
  2. \rightarrow
  3. { 0 , 1 } \{0,1\}
  4. x 1 x_{1}
  5. ¬ x 1 \lnot x_{1}
  6. x 2 x_{2}
  7. ¬ x 2 \lnot x_{2}
  8. f ( x 2 ¬ x 1 x 1 ¬ x 2 ) = ¬ x 1 f(x_{2}\leq\lnot x_{1}\leq x_{1}\leq\lnot x_{2})=\lnot x_{1}

Fuzzy_set.html

  1. ( U , m ) (U,m)
  2. U U
  3. m : U [ 0 , 1 ] . m\colon U\rightarrow[0,1].
  4. x U , x\in U,
  5. m ( x ) m(x)
  6. x x
  7. ( U , m ) . (U,m).
  8. U = { x 1 , , x n } , U=\{x_{1},\dots,x_{n}\},
  9. ( U , m ) (U,m)
  10. { m ( x 1 ) / x 1 , , m ( x n ) / x n } . \{m(x_{1})/x_{1},\dots,m(x_{n})/x_{n}\}.
  11. x U . x\in U.
  12. x x
  13. ( U , m ) (U,m)
  14. x x
  15. x x
  16. { x U m ( x ) > 0 } \{x\in U\mid m(x)>0\}
  17. ( U , m ) (U,m)
  18. { x U m ( x ) = 1 } \{x\in U\mid m(x)=1\}
  19. m m
  20. ( U , m ) . (U,m).
  21. L L
  22. L L
  23. μ : V o W \mu:\mathit{V}_{o}\to\mathit{W}
  24. V o \mathit{V}_{o}
  25. W \mathit{W}
  26. A ~ \tilde{\mathit{A}}\subseteq\mathbb{R}
  27. μ A ( x ) = 1 \mu_{A}(x)=1
  28. A ~ \tilde{\mathit{A}}\subseteq\mathbb{R}
  29. μ A ( x ) = 1 \mu_{A}(x)=1
  30. C r Cr
  31. C r { A } = 1 Cr\{A\}=1
  32. C r { B } C r { C } Cr\{B\}\leq Cr\{C\}
  33. C r { B } + C r { B c } = 1 Cr\{B\}+Cr\{B^{c}\}=1
  34. C r { A i } = sup i ( C r ( A i ) ) Cr\{\cup A_{i}\}=\sup_{i}(Cr(A_{i}))
  35. A i A_{i}
  36. sup i C r { A i } < 0.5 \sup_{i}Cr\{A_{i}\}<0.5
  37. C r { A B } = 1 2 ( sup t B u ( t ) + 1 - sup t B c u ( t ) ) Cr\{A\in B\}=\dfrac{1}{2}\left(\sup_{t\in B}u(t)+1-\sup_{t\in B^{c}}u(t)\right)
  38. E [ A ] = 0 C r { A t } d t - - 0 C r { A t } d t . E[A]=\int_{0}^{\infty}Cr\{A\geq t\}\,dt-\int_{-\infty}^{0}Cr\{A\leq t\}\,dt.
  39. H [ A ] = - S ( C r { A t } ) d t . H[A]=\int_{-\infty}^{\infty}S(Cr\{A\geq t\})\,dt.
  40. S ( y ) = - y ln y - ( 1 - y ) ln ( 1 - y ) S(y)=-y\,\,\text{ln}y-(1-y)\,\,\text{ln}(1-y)

Gabriel_Lamé.html

  1. | x a | n + | y b | n = 1 \left|\,{x\over a}\,\right|^{n}+\left|\,{y\over b}\,\right|^{n}=1

Gain.html

  1. P in / 4 π P\text{in}/4\pi
  2. gain = 10 log ( P out P in ) dB , \,\text{gain}=10\log\left(\frac{P\text{out}}{P\text{in}}\right)~{}\,\text{dB},
  3. P in P\text{in}
  4. P out P\text{out}
  5. gain = ln ( P out P in ) Np . \,\text{gain}=\ln\left(\frac{P\text{out}}{P\text{in}}\right)~{}\,\text{Np}.
  6. d b = 10 1 / 10 = 1.258925 db=10^{1/10}=1.258925\ldots
  7. log d b ( 1000 ) = 30 \log_{db}(1000)=30
  8. 10 log 10 ( 1000 ) = 10 3 = 30 10\cdot\log_{10}(1000)=10\cdot 3=30
  9. P = V 2 / R P=V^{2}/R
  10. gain = 10 log V out 2 R out V in 2 R in dB . \,\text{gain}=10\log{\frac{\frac{V\text{out}^{2}}{R\text{out}}}{\frac{V\text{% in}^{2}}{R\text{in}}}}~{}\mathrm{dB}.
  11. R in R\text{in}
  12. R out R\text{out}
  13. gain = 10 log ( V out V in ) 2 dB , \,\text{gain}=10\log\left(\frac{V\text{out}}{V\text{in}}\right)^{2}~{}\,\text{% dB},
  14. gain = 20 log ( V out V in ) dB . \,\text{gain}=20\log\left(\frac{V\text{out}}{V\text{in}}\right)~{}\,\text{dB}.
  15. P = I 2 R P=I^{2}R
  16. gain = 10 log ( I out 2 R out I in 2 R in ) dB . \,\text{gain}=10\log{\left(\frac{I\text{out}^{2}R\text{out}}{I\text{in}^{2}R% \text{in}}\right)}~{}\,\text{dB}.
  17. gain = 10 log ( I out I in ) 2 dB , \,\text{gain}=10\log\left(\frac{I\text{out}}{I\text{in}}\right)^{2}~{}\,\text{% dB},
  18. gain = 20 log ( I out I in ) dB . \,\text{gain}=20\log\left(\frac{I\text{out}}{I\text{in}}\right)~{}\,\text{dB}.
  19. h FE h\text{FE}
  20. h fe h\text{fe}
  21. I c I\text{c}
  22. I b I\text{b}
  23. I c I\text{c}
  24. I b I\text{b}
  25. h fe h\text{fe}
  26. V in V\text{in}
  27. V out V\text{out}
  28. V out V in = 10 1 = 10 V/V . \frac{V\text{out}}{V\text{in}}=\frac{10}{1}=10~{}\,\text{V/V}.
  29. V out 2 / 50 V in 2 / 50 = V out 2 V in 2 = 10 2 1 2 = 100 W/W . \frac{V\text{out}^{2}/50}{V\text{in}^{2}/50}=\frac{V\text{out}^{2}}{V\text{in}% ^{2}}=\frac{10^{2}}{1^{2}}=100~{}\,\text{W/W}.
  30. G dB = 10 log G W/W = 10 log 100 = 10 × 2 = 20 dB . G\text{dB}=10\log G\text{W/W}=10\log 100=10\times 2=20~{}\,\text{dB}.

Gall–Peters_projection.html

  1. x \displaystyle x
  2. π \pi
  3. x \displaystyle x

Galois_group.html

  1. 𝐅 q n \mathbf{F}_{q^{n}}
  2. Gal ( 𝐅 q n / 𝐅 q ) \operatorname{Gal}(\mathbf{F}_{q^{n}}/\mathbf{F}_{q})

Galois_theory.html

  1. x 2 - 4 x + 1 = 0. x^{2}-4x+1=0.
  2. A = 2 + 3 , A=2+\sqrt{3},
  3. B = 2 - 3 . B=2-\sqrt{3}.
  4. A + B = 4 , A+B=4,
  5. A B = 1. AB=1.
  6. A - B - 2 3 = 0 A-B-2\sqrt{3}=0
  7. - 2 3 -2\sqrt{3}
  8. x 4 - 10 x 2 + 1 , x^{4}-10x^{2}+1,
  9. ( x 2 - 5 ) 2 - 24. (x^{2}-5)^{2}-24.
  10. A = 2 + 3 A=\sqrt{2}+\sqrt{3}
  11. B = 2 - 3 B=\sqrt{2}-\sqrt{3}
  12. C = - 2 + 3 C=-\sqrt{2}+\sqrt{3}
  13. D = - 2 - 3 . D=-\sqrt{2}-\sqrt{3}.
  14. A B = - 1 AB=-1
  15. A C = 1 AC=1
  16. A + D = 0 A+D=0
  17. φ \varphi
  18. φ ( B ) = - 1 φ ( A ) , φ ( C ) = 1 φ ( A ) , φ ( D ) = - φ ( A ) . \varphi(B)=\frac{-1}{\varphi(A)},\quad\varphi(C)=\frac{1}{\varphi(A)},\quad% \varphi(D)=-\varphi(A).
  19. f ( x ) = x 5 - x - 1 f(x)=x^{5}-x-1
  20. x x
  21. f ( x ) = x 5 - x - 1 f(x)=x^{5}-x-1
  22. f ( x ) f(x)
  23. f ( x ) f(x)
  24. x 2 + x + 1 x^{2}+x+1
  25. f ( x ) f(x)
  26. S 5 S_{5}
  27. f ( x ) f(x)

Galvanometer.html

  1. B B
  2. B = μ 0 n I 2 r B={\mu_{0}nI\over 2r}\,
  3. I I
  4. n n
  5. r r
  6. θ θ
  7. θ = tan - 1 B B H \theta=\tan^{-1}\frac{B}{B_{H}}\,
  8. μ 0 n I 2 r = B H tan θ {\mu_{0}nI\over 2r}=B_{H}\tan\theta\,
  9. I = ( 2 r B H μ 0 n ) tan θ I=\left(\frac{2rB_{H}}{\mu_{0}n}\right)\tan\theta\,
  10. I = K t a n θ I=Ktanθ
  11. K K
  12. θ θ
  13. θ θ

Gambler's_fallacy.html

  1. 1 / 2 {1}/{2}
  2. 1 / 4 {1}/{4}
  3. 1 / 8 {1}/{8}
  4. Pr ( i = 1 n A i ) = i = 1 n Pr ( A i ) = 1 2 n \Pr\left(\bigcap_{i=1}^{n}A_{i}\right)=\prod_{i=1}^{n}\Pr(A_{i})={1\over 2^{n}}
  5. 1 / 32 {1}/{32}
  6. 1 / 32 {1}/{32}
  7. Pr ( A 5 | A 1 A 2 A 3 A 4 ) = Pr ( A 5 ) = 1 2 \Pr\left(A_{5}|A_{1}\cap A_{2}\cap A_{3}\cap A_{4}\right)=\Pr\left(A_{5}\right% )=\frac{1}{2}
  8. 1 / 32 {1}/{32}
  9. 1 / 2 {1}/{2}
  10. 1 / 2 {1}/{2}
  11. 1 - [ 15 16 ] 16 = 64.39 % 1-\left[\frac{15}{16}\right]^{16}\,=\,64.39\%
  12. 15 / 16 {15}/{16}
  13. 1 - [ 15 16 ] 15 = 62.02 % 1-\left[\frac{15}{16}\right]^{15}\,=\,62.02\%
  14. 4 / 52 {4}/{52}
  15. 3 / 51 {3}/{51}
  16. 4 / 52 {4}/{52}
  17. 4 / 51 {4}/{51}

Gamma.html

  1. γ \gamma
  2. χ \chi
  3. γ ˙ \dot{\gamma}
  4. Γ \Gamma
  5. Γ \Gamma

Gamma_correction.html

  1. V out = A V in γ V_{\,\text{out}}=A{V_{\,\text{in}}}^{\gamma}
  2. A A
  3. A = 1 A=1
  4. γ = d log ( V out ) d log ( V in ) \gamma=\frac{\mathrm{d}\log(V_{\,\text{out}})}{\mathrm{d}\log(V_{\,\text{in}})}
  5. I V S γ I\propto V_{\rm S}{}^{\gamma}
  6. V C V S ( 1 / γ ) V_{\rm C}\propto V_{\rm S}{}^{(1/\gamma)}

Gamma_function.html

  1. Γ ( n ) = ( n - 1 ) ! . \Gamma(n)=(n-1)!.
  2. Γ ( t ) = 0 x t - 1 e - x d x . \Gamma(t)=\int_{0}^{\infty}x^{t-1}e^{-x}\,dx.
  3. Γ ( t ) = { e - x } ( t ) . \Gamma(t)=\{\mathcal{M}e^{-x}\}(t).
  4. π \pi
  5. f ( 1 ) = 1 , and f ( x + 1 ) = x f ( x ) , \begin{aligned}\displaystyle f(1)&\displaystyle=1\ \,\text{, and}\\ \displaystyle f(x+1)&\displaystyle=xf(x),\end{aligned}
  6. Γ ( t ) = 0 x t - 1 e - x d x . \Gamma(t)=\int_{0}^{\infty}x^{t-1}e^{-x}\,dx.
  7. Γ ( t + 1 ) = t Γ ( t ) . \Gamma(t+1)=t\Gamma(t).
  8. Γ ( n ) = 1 2 3 ( n - 1 ) = ( n - 1 ) ! \Gamma(n)=1\cdot 2\cdot 3\cdots(n-1)=(n-1)!\,
  9. Γ ( t ) \displaystyle\Gamma(t)
  10. Γ ( t ) = x t n = 0 L n ( t ) ( x ) t + n , \Gamma(t)=x^{t}\sum_{n=0}^{\infty}\frac{L_{n}^{(t)}(x)}{t+n},
  11. | Γ ( z ) | |\Gamma(z)|
  12. Γ ( t + 1 ) 2 π t ( t e ) t , \Gamma(t+1)\sim\sqrt{2\pi t}\left(\frac{t}{e}\right)^{t},
  13. Γ ( t ) = Γ ( t + n ) t ( t + 1 ) ( t + n - 1 ) , \Gamma(t)=\frac{\Gamma(t+n)}{t(t+1)\cdots(t+n-1)},
  14. Res ( Γ , - n ) = ( - 1 ) n n ! . \operatorname{Res}(\Gamma,-n)=\frac{(-1)^{n}}{n!}.
  15. x min 1.46163 x_{\mathrm{min}}\approx 1.46163
  16. Γ ( x min ) 0.885603 \Gamma(x_{\mathrm{min}})\approx 0.885603
  17. Γ ( 1 - z ) Γ ( z ) = π sin ( π z ) , 0 < z < 1 \Gamma(1-z)\Gamma(z)={\pi\over\sin{(\pi z)}},0<z<1
  18. Γ ( ε - n ) = ( - 1 ) n - 1 Γ ( - ε ) Γ ( 1 + ε ) Γ ( n + 1 - ε ) , \Gamma(\varepsilon-n)=(-1)^{n-1}\;\frac{\Gamma(-\varepsilon)\Gamma(1+% \varepsilon)}{\Gamma(n+1-\varepsilon)},
  19. Γ ( z ) Γ ( z + 1 2 ) = 2 1 - 2 z π Γ ( 2 z ) . \Gamma(z)\Gamma\left(z+\tfrac{1}{2}\right)=2^{1-2z}\;\sqrt{\pi}\;\Gamma(2z).
  20. k = 0 m - 1 Γ ( z + k m ) = ( 2 π ) m - 1 2 m 1 2 - m z Γ ( m z ) . \prod_{k=0}^{m-1}\Gamma\left(z+\frac{k}{m}\right)=(2\pi)^{\frac{m-1}{2}}\;m^{% \frac{1}{2}-mz}\;\Gamma(mz).
  21. Γ ( z ) ¯ = Γ ( z ¯ ) Γ ( z ) Γ ( z ¯ ) 𝐑 . \overline{\Gamma(z)}=\Gamma(\overline{z})\;\Rightarrow\;\Gamma(z)\Gamma(% \overline{z})\in\mathbf{R}.
  22. Γ ( 1 2 ) = π , \Gamma\left(\tfrac{1}{2}\right)=\sqrt{\pi},
  23. Γ ( 1 2 + n ) \displaystyle\Gamma\left(\tfrac{1}{2}+n\right)
  24. lim n Γ ( n + α ) Γ ( n ) n α = 1 , α 𝐑 \lim_{n\to\infty}\frac{\Gamma(n+\alpha)}{\Gamma(n)n^{\alpha}}=1,\qquad\alpha% \in\mathbf{R}
  25. Γ ( z ) = Γ ( z ) ψ 0 ( z ) . \Gamma^{\prime}(z)=\Gamma(z)\psi_{0}(z).
  26. Γ ( m + 1 ) = m ! ( - γ + k = 1 m 1 k ) . \Gamma^{\prime}(m+1)=m!\left(-\gamma+\sum_{k=1}^{m}\frac{1}{k}\right)\,.
  27. d n d x n Γ ( x ) = 0 t x - 1 e - t ( ln t ) n d t . \frac{{\rm d}^{n}}{{\rm d}x^{n}}\,\Gamma(x)=\int_{0}^{\infty}t^{x-1}e^{-t}(\ln t% )^{n}dt.
  28. Res ( Γ , - n ) = ( - 1 ) n n ! . \operatorname{Res}(\Gamma,-n)=\frac{(-1)^{n}}{n!}.
  29. Γ ( z ) = n = 0 ( - 1 ) n n ! 1 z + n + 1 t z - 1 e - t d t . \Gamma(z)=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\frac{1}{z+n}+\int_{1}^{\infty% }t^{z-1}e^{-t}dt.
  30. Γ ( z ) = 1 + n = 1 Γ ( n ) ( 1 ) n ! ( z - 1 ) n , \Gamma(z)=1+\sum_{n=1}^{\infty}\frac{\Gamma^{(n)}(1)}{n!}(z-1)^{n},
  31. π = ( a 1 , , a 1 k 1 , , a r , , a r k r ) \pi=(\underbrace{a_{1},\dots,a_{1}}_{k_{1}},\dots,\underbrace{a_{r},\dots,a_{r% }}_{k_{r}})
  32. ζ * ( x ) := { ζ ( x ) x 1 γ x = 1 \zeta^{*}(x):=\begin{cases}\zeta(x)&x\neq 1\\ \gamma&x=1\end{cases}
  33. Γ ( z ) = 1 z - γ + 1 2 ( γ 2 + π 2 6 ) z - 1 6 ( γ 3 + γ π 2 2 + 2 ζ ( 3 ) ) z 2 + O ( z 3 ) \Gamma(z)=\frac{1}{z}-\gamma+\frac{1}{2}\left(\gamma^{2}+\frac{\pi^{2}}{6}% \right)z-\frac{1}{6}\left(\gamma^{3}+\frac{\gamma\pi^{2}}{2}+2\zeta(3)\right)z% ^{2}+O(z^{3})
  34. Γ ( z ) \Gamma(z)
  35. ln Γ ( 1 + z ) = - γ z + k = 2 ζ ( k ) k ( - z ) k | z | < 1 \ln\Gamma(1+z)=-\gamma z+\sum_{k=2}^{\infty}\frac{\zeta(k)}{k}\,(-z)^{k}\qquad% \forall\;|z|<1
  36. ζ ( s ) Γ ( s ) = 0 t s e t - 1 d t t \zeta(s)\Gamma(s)=\int_{0}^{\infty}\frac{t^{s}}{e^{t}-1}\;\frac{dt}{t}
  37. ln Γ ( 1 + z ) = - γ z + 0 e - z t - 1 + z t t ( e t - 1 ) d t \ln\Gamma(1+z)=-\gamma z+\int_{0}^{\infty}\frac{e^{-zt}-1+zt}{t(e^{t}-1)}dt
  38. γ \gamma
  39. ln Γ ( 1 + z ) = 0 e - z t - z e - t - 1 + z t ( e t - 1 ) d t \ln\Gamma(1+z)=\int_{0}^{\infty}\frac{e^{-zt}-ze^{-t}-1+z}{t(e^{t}-1)}dt
  40. ln Γ ( x ) = ( 1 2 - x ) ( γ + ln 2 ) + ( 1 - x ) ln π - 1 2 ln sin π x + 1 π n = 1 sin 2 π n x ln n n , 0 < x < 1 , \ln\Gamma(x)=\left(\frac{1}{2}-x\right)(\gamma+\ln 2)+(1-x)\ln\pi-\frac{1}{2}% \ln\sin\pi x+\frac{1}{\pi}\sum_{n=1}^{\infty}\frac{\sin 2\pi nx\cdot\ln{n}}{n}% \,,\qquad 0<x<1,
  41. a a + 1 ln Γ ( t ) d t = 1 2 ln 2 π + a ln a - a , a > 0. \int\limits_{a}^{a+1}\ln\Gamma(t)\,dt=\tfrac{1}{2}\ln 2\pi+a\ln a-a,\quad a>0.
  42. a = 0 a=0
  43. 0 1 ln Γ ( t ) d t = 1 2 ln 2 π . \int\limits_{0}^{1}\ln\Gamma(t)\,dt=\tfrac{1}{2}\ln 2\pi.
  44. ln Γ ( 1 + x ) = x ln x - x + 1 2 ln 2 π x + 0 2 a r c t a n ( t x ) e 2 π t - 1 d t \ln\Gamma(1+x)=x\ln x-x+\frac{1}{2}\ln 2\pi x+\int\limits_{0}^{\infty}\frac{2% \mathrm{arctan}(\frac{t}{x})}{e^{2\pi t}-1}dt
  45. Π ( z ) = Γ ( z + 1 ) = z Γ ( z ) = 0 e - t t z d t , \Pi(z)=\Gamma(z+1)=z\Gamma(z)=\int_{0}^{\infty}e^{-t}t^{z}\,dt,
  46. Π ( n ) = n ! , \Pi(n)=n!,
  47. Π ( z ) Π ( - z ) = π z sin ( π z ) = 1 sinc ( z ) \Pi(z)\Pi(-z)=\frac{\pi z}{\sin(\pi z)}=\frac{1}{\operatorname{sinc}(z)}
  48. Π ( z m ) Π ( z - 1 m ) Π ( z - m + 1 m ) = ( 2 π ) m - 1 2 m - z - 1 2 Π ( z ) . \Pi\left(\frac{z}{m}\right)\,\Pi\left(\frac{z-1}{m}\right)\cdots\Pi\left(\frac% {z-m+1}{m}\right)=(2\pi)^{\frac{m-1}{2}}m^{-z-\frac{1}{2}}\Pi(z).
  49. π ( z ) = 1 Π ( z ) , \pi(z)=\frac{1}{\Pi(z)},
  50. r 1 , , r n r_{1},\ldots,r_{n}
  51. V n ( r 1 , , r n ) = π n 2 Π ( n 2 ) k = 1 n r k . V_{n}(r_{1},\ldots,r_{n})=\frac{\pi^{\frac{n}{2}}}{\Pi\left(\frac{n}{2}\right)% }\prod_{k=1}^{n}r_{k}.
  52. 0 z log Γ ( x ) d x \int_{0}^{z}\log\Gamma(x)\ \mathrm{d}x
  53. 0 z log Γ ( x ) d x = z 2 log ( 2 π ) + z ( 1 - z ) 2 + z log Γ ( z ) - log G ( z + 1 ) \int_{0}^{z}\log\Gamma(x)\ \mathrm{d}x=\frac{z}{2}\log(2\pi)+\frac{z(1-z)}{2}+% z\log\Gamma(z)-\log G(z+1)
  54. ( z ) > - 1 \Re(z)>-1
  55. n / 4 m / 6 log F ( z ) d z \int_{n/4}^{m/6}\log F(z)\,\text{d}z
  56. 0 z log Γ ( x ) d x = z 2 log ( 2 π ) + z ( 1 - z ) 2 - ζ ( - 1 ) + ζ ( - 1 , z ) . \int_{0}^{z}\log\Gamma(x)\ \mathrm{d}x=\frac{z}{2}\log(2\pi)+\frac{z(1-z)}{2}-% \zeta^{{}^{\prime}}(-1)+\zeta^{{}^{\prime}}(-1,z).
  57. B ( x , y ) = Γ ( x ) Γ ( y ) Γ ( x + y ) . B(x,y)=\frac{\Gamma(x)\;\Gamma(y)}{\Gamma(x+y)}.
  58. π - z 2 Γ ( z 2 ) ζ ( z ) = π - 1 - z 2 Γ ( 1 - z 2 ) ζ ( 1 - z ) . \pi^{-\frac{z}{2}}\;\Gamma\left(\frac{z}{2}\right)\zeta(z)=\pi^{-\frac{1-z}{2}% }\;\Gamma\left(\frac{1-z}{2}\right)\;\zeta(1-z).
  59. ζ ( z ) Γ ( z ) = 0 u z e u - 1 d u u , \zeta(z)\Gamma(z)=\int_{0}^{\infty}\frac{u^{z}}{e^{u}-1}\;\frac{du}{u},
  60. log Γ ( x ) = ζ H ( 0 , x ) - ζ ( 0 ) , \log\Gamma(x)=\zeta_{H}^{\prime}(0,x)-\zeta^{\prime}(0),
  61. τ n 0 d t t n - 1 e - ( t τ ) β = τ n β Γ ( n β ) . \langle\tau^{n}\rangle\equiv\int_{0}^{\infty}dt\;t^{n-1}\,e^{-\left(\frac{t}{% \tau}\right)^{\beta}}=\frac{\tau^{n}}{\beta}\Gamma\left({n\over\beta}\right).
  62. Γ ( - 1 ) \displaystyle\Gamma(-1)
  63. Γ ( z ) = 0 x e - t t z - 1 d t t + x e - t t z + 1 d t t = x z e - x n = 0 x n z ( z + 1 ) ( z + n ) + x e - t t z d t t . \begin{aligned}\displaystyle\Gamma(z)&\displaystyle=\int_{0}^{x}e^{-t}t^{z-1}% \,\frac{dt}{t}+\int_{x}^{\infty}e^{-t}t^{z+1}\,\frac{dt}{t}\\ &\displaystyle=x^{z}e^{-x}\sum_{n=0}^{\infty}\frac{x^{n}}{z(z+1)\cdots(z+n)}+% \int_{x}^{\infty}e^{-t}t^{z}\,\frac{dt}{t}.\end{aligned}
  64. ln ( Γ ( z ) ) = ln ( Γ ( z + 1 ) ) - ln ( z ) \ln(\Gamma(z))=\ln(\Gamma(z+1))-\ln(z)
  65. ln ( Γ ( z ) ) ( z - 1 2 ) ln ( z ) - z + 1 2 ln ( 2 π ) . \ln(\Gamma(z))\approx(z-\tfrac{1}{2})\ln(z)-z+\tfrac{1}{2}\ln(2\pi).
  66. ln ( Γ ( z - m ) ) = ln ( Γ ( z ) ) - k = 1 m ln ( z - k ) . \ln(\Gamma(z-m))=\ln(\Gamma(z))-\sum_{k=1}^{m}\ln(z-k).
  67. Γ ( z ) z z - 1 2 e - z 2 π ( 1 + 1 12 z + 1 288 z 2 - 139 51840 z 3 - 571 2488320 z 4 ) \Gamma(z)\sim z^{z-\frac{1}{2}}e^{-z}\sqrt{2\pi}\left(1+\frac{1}{12z}+\frac{1}% {288z^{2}}-\frac{139}{51840z^{3}}-\frac{571}{2488320z^{4}}\right)
  68. as | z | at constant | arg ( z ) | < π \,\text{as }|z|\to\infty\,\text{ at constant}\quad|\arg(z)|<\pi
  69. ln Γ ( z ) z ln ( z ) - z - 1 2 ln ( z 2 π ) + 1 12 z - 1 360 z 3 + 1 1260 z 5 \ln\Gamma(z)\sim z\ln(z)-z-\tfrac{1}{2}\ln\left(\frac{z}{2\pi}\right)+\frac{1}% {12z}-\frac{1}{360z^{3}}+\frac{1}{1260z^{5}}
  70. as | z | at constant | arg ( z ) | < π \,\text{as }|z|\to\infty\,\text{ at constant}\quad|\arg(z)|<\pi
  71. B k k ( k - 1 ) \frac{B_{k}}{k(k-1)}
  72. f ( t ) e - g ( t ) f(t)\,e^{-g(t)}
  73. 0 t b e - a t d t = Γ ( b + 1 ) a b + 1 . \int_{0}^{\infty}t^{b}e^{-at}\,dt=\frac{\Gamma(b+1)}{a^{b+1}}.
  74. a e - ( x - b ) 2 c 2 ae^{-\frac{(x-b)^{2}}{c^{2}}}
  75. B ( x , y ) = 0 1 t x - 1 ( 1 - t ) y - 1 d t = Γ ( x ) Γ ( y ) Γ ( x + y ) . \mathrm{B}(x,y)=\int_{0}^{1}t^{x-1}(1-t)^{y-1}\,dt=\frac{\Gamma(x)\,\Gamma(y)}% {\Gamma(x+y)}.
  76. ( n k ) = n ! k ! ( n - k ) ! . {n\choose k}=\frac{n!}{k!(n-k)!}.
  77. p 1 p m p_{1}\ldots p_{m}
  78. q 1 q n q_{1}\ldots q_{n}
  79. i = a b P ( i ) Q ( i ) = ( j = 1 m Γ ( b - p j + 1 ) Γ ( a - p j ) ) ( k = 1 n Γ ( a - q k ) Γ ( b - q k + 1 ) ) . \prod_{i=a}^{b}\frac{P(i)}{Q(i)}=\left(\prod_{j=1}^{m}\frac{\Gamma(b-p_{j}+1)}% {\Gamma(a-p_{j})}\right)\left(\prod_{k=1}^{n}\frac{\Gamma(a-q_{k})}{\Gamma(b-q% _{k}+1)}\right).
  80. Γ ( s 2 ) ζ ( s ) π - s 2 = Γ ( 1 - s 2 ) ζ ( 1 - s ) π - 1 - s 2 . \Gamma\left(\frac{s}{2}\right)\zeta(s)\pi^{-\frac{s}{2}}=\Gamma\left(\frac{1-s% }{2}\right)\zeta(1-s)\pi^{-\frac{1-s}{2}}.
  81. ζ ( s ) Γ ( s ) = 0 t s e t - 1 d t t . \zeta(s)\;\Gamma(s)=\int_{0}^{\infty}\frac{t^{s}}{e^{t}-1}\;\frac{dt}{t}.
  82. n ! = k = 1 ( 1 + 1 k ) n 1 + n k , n!=\prod_{k=1}^{\infty}\frac{\left(1+\frac{1}{k}\right)^{n}}{1+\frac{n}{k}}\,,
  83. n ! = 0 1 ( - ln s ) n d s , n!=\int_{0}^{1}(-\ln s)^{n}\,ds\,,
  84. Γ ( z ) = lim m m z m ! z ( z + 1 ) ( z + 2 ) ( z + m ) \Gamma(z)=\lim_{m\to\infty}\frac{m^{z}m!}{z(z+1)(z+2)\cdots(z+m)}
  85. Γ ( z ) = e - γ z z k = 1 ( 1 + z k ) - 1 e z k , \Gamma(z)=\frac{e^{-\gamma z}}{z}\prod_{k=1}^{\infty}\left(1+\frac{z}{k}\right% )^{-1}e^{\frac{z}{k}},

Garnet.html

  1. 3 ¯ \overline{3}
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  50. 3 ¯ \overline{3}

Gas_constant.html

  1. R R
  2. R ¯ \overline{R}
  3. R = 8.314 4621 ( 75 ) J mol K R=8.314\,4621(75)~{}\frac{\mathrm{J}}{\mathrm{mol}\cdot\mathrm{K}}
  4. × 10 7 \times 10^{−}7
  5. P V = N R T = m R specific T PV=NRT=mR_{\rm specific}T\,\!
  6. R = P V N T R=\frac{PV}{NT}
  7. R = force / area × volume amount × temperature R=\frac{\mathrm{force/area}\times\mathrm{volume}}{\mathrm{amount}\times\mathrm% {temperature}}
  8. R = force / ( length ) 2 × ( length ) 3 amount × temperature R=\frac{\mathrm{force}/(\mathrm{length})^{2}\times(\mathrm{length})^{3}}{% \mathrm{amount}\times\mathrm{temperature}}
  9. R = work amount × temperature R=\frac{\mathrm{work}}{\mathrm{amount}\times\mathrm{temperature}}
  10. R = N A k B , \qquad R=N_{\rm A}k_{\rm B},\,
  11. P V = k B N T . PV=k_{\rm B}NT.\,\!
  12. P = k B n T . P=k_{\rm B}nT.\,\!
  13. c a 2 ( 0 , T ) = γ 0 R T A r ( Ar ) M u , c_{\mathrm{a}}^{2}(0,T)=\frac{\gamma_{0}RT}{A_{\mathrm{r}}(\mathrm{Ar})M_{% \mathrm{u}}},
  14. R specific = R M R_{\rm specific}=\frac{R}{M}
  15. R specific = k B m R_{\rm specific}=\frac{{k_{\rm B}}}{m}
  16. R specific = c p - c v R_{\rm specific}=c_{\rm p}-c_{\rm v}
  17. R * = 8.314 32 N m kmol K . R^{*}=8.314\,32\frac{\mathrm{N\,m}}{\mathrm{kmol\,K}}.

Gas_laws.html

  1. P 1 V P\propto\frac{1}{V}
  2. P V = k 1 PV=k_{1}
  3. P 1 V 1 = P 2 V 2 P_{1}V_{1}=P_{2}V_{2}\,
  4. V T V\propto T\,
  5. V / T = k 2 V/T=k_{2}
  6. V 1 / T 1 = V 2 / T 2 V_{1}/T_{1}=V_{2}/T_{2}
  7. P T P\propto T\,
  8. P / T = k 3 P/T=k_{3}
  9. P 1 / T 1 = P 2 / T 2 P_{1}/T_{1}=P_{2}/T_{2}
  10. V 1 n 1 = V 2 n 2 \frac{V_{1}}{n_{1}}=\frac{V_{2}}{n_{2}}\,
  11. p V = k 5 T pV=k_{5}T\,
  12. p 1 V 1 T 1 = p 2 V 2 T 2 \qquad\frac{p_{1}V_{1}}{T_{1}}=\frac{p_{2}V_{2}}{T_{2}}
  13. p V = n R T pV=nRT\,
  14. p V = k N T pV=kNT\,
  15. P t o t a l = P 1 + P 2 + P 3 + + P n i = 1 n P i P_{total}=P_{1}+P_{2}+P_{3}+...+P_{n}\equiv\sum_{i=1}^{n}P_{i}\,
  16. P total = P gas + P H 2 O P_{\mathrm{total}}=P_{\mathrm{gas}}+P_{\mathrm{H_{2}O}}\,
  17. p = k H c p=k_{\rm H}\,c

Gateway_Arch.html

  1. y = A ( cosh C x L - 1 ) x = L C cosh - 1 ( 1 + y A ) y=A\left(\cosh\frac{Cx}{L}-1\right)\quad\Leftrightarrow\quad x=\frac{L}{C}% \cosh^{-1}\left(1+\frac{y}{A}\right)
  2. A = f c Q b / Q t - 1 = 68.7672 A=\frac{f_{c}}{Q_{b}/Q_{t}-1}=68.7672
  3. C = cosh - 1 Q b Q t = 3.0022 C=\cosh^{-1}\frac{Q_{b}}{Q_{t}}=3.0022

Gaussian_beam.html

  1. E ( r , z ) = E 0 w 0 w ( z ) exp ( - r 2 w ( z ) 2 - i k z - i k r 2 2 R ( z ) + i ζ ( z ) ) , E(r,z)=E_{0}\frac{w_{0}}{w(z)}\exp\left(\frac{-r^{2}}{w(z)^{2}}-ikz-ik\frac{r^% {2}}{2R(z)}+i\zeta(z)\right)\ ,
  2. r r
  3. z z
  4. i i
  5. i 2 = - 1 i^{2}=-1
  6. k = 2 π / λ k=2\pi/\lambda
  7. E 0 = | E ( 0 , 0 ) | E_{0}=|E(0,0)|
  8. w ( z ) w(z)
  9. w 0 = w ( 0 ) w_{0}=w(0)
  10. R ( z ) R(z)
  11. ζ ( z ) \zeta(z)
  12. e i ω t e^{i\omega t}
  13. I ( r , z ) = | E ( r , z ) | 2 2 η = I 0 ( w 0 w ( z ) ) 2 exp ( - 2 r 2 w 2 ( z ) ) , I(r,z)={|E(r,z)|^{2}\over 2\eta}=I_{0}\left(\frac{w_{0}}{w(z)}\right)^{2}\exp% \left(\frac{-2r^{2}}{w^{2}(z)}\right)\ ,
  14. I 0 = I ( 0 , 0 ) I_{0}=I(0,0)
  15. η \eta\,
  16. η = η 0 = μ 0 / ε 0 = 1 / ( ε 0 c ) 376.7 Ω \eta=\eta_{0}=\sqrt{\mu_{0}/\varepsilon_{0}}=1/(\varepsilon_{0}c)\approx 376.7\ \Omega
  17. w ( z ) = w 0 1 + ( z z R ) 2 . w(z)=w_{0}\,\sqrt{1+{\left(\frac{z}{z_{\mathrm{R}}}\right)}^{2}}\ .
  18. z R = π w 0 2 λ z_{\mathrm{R}}=\frac{\pi w_{0}^{2}}{\lambda}
  19. w ( ± z R ) = 2 w 0 . w(\pm z_{\mathrm{R}})=\sqrt{2}w_{0}.
  20. b = 2 z R = 2 π w 0 2 λ . b=2z_{\mathrm{R}}=\frac{2\pi w_{0}^{2}}{\lambda}\ .
  21. R ( z ) = z [ 1 + ( z R z ) 2 ] . R(z)=z\left[{1+{\left(\frac{z_{\mathrm{R}}}{z}\right)}^{2}}\right]\ .
  22. w ( z ) w(z)
  23. z z
  24. z z R z\gg z_{\mathrm{R}}
  25. r = w ( z ) r=w(z)
  26. r = 0 r=0
  27. θ λ π w 0 ( θ in radians ) . \theta\simeq\frac{\lambda}{\pi w_{0}}\qquad(\theta\mathrm{\ in\ radians}).
  28. Θ = 2 θ . \Theta=2\theta\ .
  29. 2 λ / π 2\lambda/\pi
  30. w 0 w_{0}
  31. NA = n sin θ \mathrm{NA}=n\sin\theta
  32. z R = w 0 / NA z_{\mathrm{R}}=w_{0}/\mathrm{NA}
  33. ζ ( z ) = arctan ( z z R ) . \zeta(z)=\arctan\left(\frac{z}{z_{\mathrm{R}}}\right)\ .
  34. e - i k z e^{-ikz}
  35. q ( z ) q(z)
  36. q ( z ) = z + q 0 = z + i z R . q(z)=z+q_{0}=z+iz_{\mathrm{R}}\ .
  37. 1 / q ( z ) 1/q(z)
  38. q ( z ) q(z)
  39. w ( z ) w(z)
  40. R ( z ) R(z)
  41. 1 q ( z ) = 1 z + i z R = z z 2 + z R 2 - i z R z 2 + z R 2 = 1 R ( z ) - i λ π w 2 ( z ) . {1\over q(z)}={1\over z+iz_{\mathrm{R}}}={z\over z^{2}+z_{\mathrm{R}}^{2}}-i{z% _{\mathrm{R}}\over z^{2}+z_{\mathrm{R}}^{2}}={1\over R(z)}-i{\lambda\over\pi w% ^{2}(z)}.
  42. u ( x , y , z ) = u ( x , z ) u ( y , z ) , {u}(x,y,z)={u}(x,z)\,{u}(y,z),
  43. u ( x , z ) = 1 q x ( z ) exp ( - i k x 2 2 q x ( z ) ) {u}(x,z)=\frac{1}{\sqrt{{q}_{x}(z)}}\exp\left(-ik\frac{x^{2}}{2{q}_{x}(z)}\right)
  44. u ( y , z ) = 1 q y ( z ) exp ( - i k y 2 2 q y ( z ) ) {u}(y,z)=\frac{1}{\sqrt{{q}_{y}(z)}}\exp\left(-ik\frac{y^{2}}{2{q}_{y}(z)}\right)
  45. q x ( z ) q_{x}(z)
  46. q y ( z ) q_{y}(z)
  47. q x = q y = q {q}_{x}={q}_{y}={q}
  48. x 2 + y 2 = r 2 x^{2}+y^{2}=r^{2}
  49. u ( r , z ) = 1 q ( z ) exp ( - i k r 2 2 q ( z ) ) . {u}(r,z)=\frac{1}{{q}(z)}\exp\left(-ik\frac{r^{2}}{2{q}(z)}\right).
  50. P ( r , z ) = P 0 [ 1 - e - 2 r 2 / w 2 ( z ) ] , P(r,z)=P_{0}\left[1-e^{-2r^{2}/w^{2}(z)}\right]\ ,
  51. P 0 = 1 2 π I 0 w 0 2 P_{0}={1\over 2}\pi I_{0}w_{0}^{2}
  52. r = w ( z ) r=w(z)\,
  53. P ( z ) P 0 = 1 - e - 2 0.865 . {P(z)\over P_{0}}=1-e^{-2}\approx 0.865\ .
  54. r = 1.07 w ( z ) r=1.07\cdot w(z)
  55. r = 1.224 w ( z ) r=1.224\cdot w(z)
  56. r = 1.52 w ( z ) r=1.52\cdot w(z)
  57. z z
  58. r r
  59. π r 2 \pi r^{2}
  60. I ( 0 , z ) = lim r 0 P 0 [ 1 - e - 2 r 2 / w 2 ( z ) ] π r 2 = P 0 π lim r 0 [ - ( - 2 ) ( 2 r ) e - 2 r 2 / w 2 ( z ) ] w 2 ( z ) ( 2 r ) = 2 P 0 π w 2 ( z ) . I(0,z)=\lim_{r\to 0}\frac{P_{0}\left[1-e^{-2r^{2}/w^{2}(z)}\right]}{\pi r^{2}}% =\frac{P_{0}}{\pi}\lim_{r\to 0}\frac{\left[-(-2)(2r)e^{-2r^{2}/w^{2}(z)}\right% ]}{w^{2}(z)(2r)}={2P_{0}\over\pi w^{2}(z)}.
  61. w ( z ) w(z)
  62. 2 U = 1 c 2 2 U t 2 , \nabla^{2}U=\frac{1}{c^{2}}\frac{\partial^{2}U}{\partial t^{2}},
  63. U U
  64. E x E_{x}
  65. E y E_{y}
  66. E z E_{z}
  67. B x B_{x}
  68. B y B_{y}
  69. B z B_{z}
  70. U ( x , y , z , t ) = u ( x , y , z ) e - i ( k z - ω t ) , U(x,y,z,t)=u(x,y,z)e^{-i(kz-\omega t)},
  71. z z
  72. 2 u / z 2 \partial^{2}u/\partial z^{2}
  73. 2 u x 2 + 2 u y 2 = 2 i k u z . \frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}}=2ik% \frac{\partial u}{\partial z}.
  74. q q
  75. x x
  76. u n ( x , z ) = ( 2 π ) 1 / 4 ( 1 2 n n ! w 0 ) 1 / 2 ( q 0 q ( z ) ) 1 / 2 [ q 0 q 0 q ( z ) q ( z ) ] n / 2 H n ( 2 x w ( z ) ) exp [ - i k x 2 2 q ( z ) ] {u}_{n}(x,z)=\left(\frac{2}{\pi}\right)^{1/4}\left(\frac{1}{2^{n}n!w_{0}}% \right)^{1/2}\left(\frac{{q}_{0}}{{q}(z)}\right)^{1/2}\left[\frac{{q}_{0}}{{q}% _{0}^{\ast}}\frac{{q}^{\ast}(z)}{{q}(z)}\right]^{n/2}H_{n}\left(\frac{\sqrt{2}% x}{w(z)}\right)\exp\left[-i\frac{kx^{2}}{2{q}(z)}\right]
  77. H n ( x ) H_{n}(x)
  78. n n
  79. H 1 ( x ) = 2 x H_{1}(x)=2x\,
  80. n = 0 n=0
  81. u m n ( x , y , z ) = u m ( x , z ) u n ( y , z ) {u}_{mn}(x,y,z)=u_{m}(x,z)u_{n}(y,z)
  82. u n ( y , z ) u_{n}(y,z)
  83. u m ( x , z ) u_{m}(x,z)
  84. u ( r , ϕ , z ) = C l p L G w ( z ) ( r 2 w ( z ) ) | l | exp ( - r 2 w 2 ( z ) ) L p | l | ( 2 r 2 w 2 ( z ) ) exp ( - i k r 2 2 R ( z ) ) exp ( i l ϕ ) exp [ i ( 2 p + | l | + 1 ) ζ ( z ) ] , {u}(r,\phi,z)=\frac{C^{LG}_{lp}}{w(z)}\left(\frac{r\sqrt{2}}{w(z)}\right)^{|l|% }\exp\left(-\frac{r^{2}}{w^{2}(z)}\right)L_{p}^{|l|}\left(\frac{2r^{2}}{w^{2}(% z)}\right)\exp\left(-ik\frac{r^{2}}{2R(z)}\right)\exp(il\phi)\exp\left[i(2p+|l% |+1)\zeta(z)\right],
  85. L p l L_{p}^{l}
  86. p 0 p\geq 0
  87. l l
  88. C l p L G C^{LG}_{lp}
  89. w ( z ) w(z)
  90. R ( z ) R(z)
  91. ζ ( z ) \zeta(z)
  92. u ε ( ξ , η , z ) = w 0 w ( z ) C p m ( i ξ , ε ) C p m ( η , ε ) exp [ - i k r 2 2 q ( z ) - ( p + 1 ) ζ ( z ) ] , u_{\varepsilon}\left(\xi,\eta,z\right)=\frac{w_{0}}{w\left(z\right)}\mathrm{C}% _{p}^{m}\left(i\xi,\varepsilon\right)\mathrm{C}_{p}^{m}\left(\eta,\varepsilon% \right)\exp\left[-ik\frac{r^{2}}{2q\left(z\right)}-\left(p+1\right)\zeta\left(% z\right)\right],
  93. ξ \xi
  94. η \eta
  95. x = ε / 2 w ( z ) cosh ξ cos η , x=\sqrt{\varepsilon/2}w\left(z\right)\cosh\xi\cos\eta,
  96. y = ε / 2 w ( z ) sinh ξ sin η . y=\sqrt{\varepsilon/2}w\left(z\right)\sinh\xi\sin\eta.
  97. C p m ( η , ε ) {C}_{p}^{m}\left(\eta,\varepsilon\right)
  98. p p
  99. m m
  100. ε \varepsilon
  101. ε = \varepsilon=\infty
  102. ε = 0 \varepsilon=0
  103. ρ = r / w 0 \rho=r/w_{0}
  104. \Zeta = z / z R \Zeta=z/z_{R}
  105. u p m ( ρ , θ ; \Zeta ) = 2 p + | m | + 1 π Γ ( p + | m | + 1 ) Γ ( 1 + | m | + p 2 ) Γ ( | m | + 1 ) i | m | + 1 \Zeta p 2 ( \Zeta + i ) - ( 1 + | m | + p 2 ) ρ | m | e - i ρ 2 ( \Zeta + i ) e i m ϕ F 1 1 ( - p 2 , | m | + 1 ; r 2 \Zeta ( \Zeta + i ) ) , u_{pm}(\rho,\theta;\Zeta)=\sqrt{\frac{2^{p+|m|+1}}{\pi\Gamma(p+|m|+1)}}\frac{% \Gamma(1+|m|+\frac{p}{2})}{\Gamma(|m|+1)}\,\,i^{|m|+1}\Zeta^{\frac{p}{2}}(% \Zeta+i)^{-(1+|m|+\frac{p}{2})}\rho^{|m|}e^{-\frac{i\rho^{2}}{(\Zeta+i)}}e^{im% \phi}{}_{1}F_{1}\left(-\frac{p}{2},|m|+1;\frac{r^{2}}{\Zeta(\Zeta+i)}\right),
  106. m m
  107. p - | m | p\geq-|m|
  108. Γ ( x ) \Gamma(x)
  109. F 1 1 ( a , b ; x ) {}_{1}F_{1}(a,b;x)
  110. \Zeta = 0 \Zeta=0
  111. u ( ρ , ϕ , 0 ) ρ p + | m | e - ρ 2 + i m ϕ . u(\rho,\phi,0)\propto\rho^{p+|m|}e^{-\rho^{2}+im\phi}.
  112. ζ \zeta

Gaussian_elimination.html

  1. [ 1 3 1 9 1 1 - 1 1 3 11 5 35 ] [ 1 3 1 9 0 - 2 - 2 - 8 0 2 2 8 ] [ 1 3 1 9 0 - 2 - 2 - 8 0 0 0 0 ] [ 1 0 - 2 - 3 0 1 1 4 0 0 0 0 ] \left[\begin{array}[]{rrr|r}1&3&1&9\\ 1&1&-1&1\\ 3&11&5&35\end{array}\right]\to\left[\begin{array}[]{rrr|r}1&3&1&9\\ 0&-2&-2&-8\\ 0&2&2&8\end{array}\right]\to\left[\begin{array}[]{rrr|r}1&3&1&9\\ 0&-2&-2&-8\\ 0&0&0&0\end{array}\right]\to\left[\begin{array}[]{rrr|r}1&0&-2&-3\\ 0&1&1&4\\ 0&0&0&0\end{array}\right]
  2. [ 0 \color r e d 𝟐 1 - 1 0 0 \color r e d 𝟑 1 0 0 0 0 ] \left[\begin{array}[]{cccc}0&\color{red}{\mathbf{2}}&1&-1\\ 0&0&\color{red}{\mathbf{3}}&1\\ 0&0&0&0\end{array}\right]
  3. 2 x \displaystyle 2x
  4. L 1 L_{1}
  5. L 2 L_{2}
  6. 2 x + y - z = 8 - 3 x - y + 2 z = - 11 - 2 x + y + 2 z = - 3 \begin{aligned}\displaystyle 2x&&\displaystyle\;+&&\displaystyle y&&% \displaystyle\;-&&\displaystyle z&&\displaystyle\;=&&\displaystyle 8&\\ \displaystyle-3x&&\displaystyle\;-&&\displaystyle y&&\displaystyle\;+&&% \displaystyle 2z&&\displaystyle\;=&&\displaystyle-11&\\ \displaystyle-2x&&\displaystyle\;+&&\displaystyle y&&\displaystyle\;+&&% \displaystyle 2z&&\displaystyle\;=&&\displaystyle-3&\end{aligned}
  7. [ 2 1 - 1 8 - 3 - 1 2 - 11 - 2 1 2 - 3 ] \left[\begin{array}[]{ccc|c}2&1&-1&8\\ -3&-1&2&-11\\ -2&1&2&-3\end{array}\right]
  8. 2 x + y - z = 8 1 2 y + 1 2 z = 1 2 y + z = 5 \begin{aligned}\displaystyle 2x&&\displaystyle\;+&&\displaystyle y&&% \displaystyle\;-&&\displaystyle\;z&&\displaystyle\;=&&\displaystyle 8&\\ &&&&\displaystyle\frac{1}{2}y&&\displaystyle\;+&&\displaystyle\;\frac{1}{2}z&&% \displaystyle\;=&&\displaystyle 1&\\ &&&&\displaystyle 2y&&\displaystyle\;+&&\displaystyle\;z&&\displaystyle\;=&&% \displaystyle 5&\end{aligned}
  9. L 2 + 3 2 L 1 L 2 L_{2}+\frac{3}{2}L_{1}\rightarrow L_{2}
  10. L 3 + L 1 L 3 L_{3}+L_{1}\rightarrow L_{3}
  11. [ 2 1 - 1 8 0 1 / 2 1 / 2 1 0 2 1 5 ] \left[\begin{array}[]{ccc|c}2&1&-1&8\\ 0&1/2&1/2&1\\ 0&2&1&5\end{array}\right]
  12. 2 x + y - z = 8 1 2 y + 1 2 z = 1 - z = 1 \begin{aligned}\displaystyle 2x&&\displaystyle\;+&&\displaystyle y&&% \displaystyle-&&\displaystyle\;z&&\displaystyle=&&\displaystyle 8&\\ &&&&\displaystyle\frac{1}{2}y&&\displaystyle+&&\displaystyle\;\frac{1}{2}z&&% \displaystyle=&&\displaystyle 1&\\ &&&&&&&&\displaystyle\;-z&&\displaystyle\;=&&\displaystyle 1&\end{aligned}
  13. L 3 + - 4 L 2 L 3 L_{3}+-4L_{2}\rightarrow L_{3}
  14. [ 2 1 - 1 8 0 1 / 2 1 / 2 1 0 0 - 1 1 ] \left[\begin{array}[]{ccc|c}2&1&-1&8\\ 0&1/2&1/2&1\\ 0&0&-1&1\end{array}\right]
  15. 2 x + y = 7 1 2 y = 3 / 2 - z = 1 \begin{aligned}\displaystyle 2x&&\displaystyle\;+&&\displaystyle y&&&&&&% \displaystyle=&&\displaystyle 7&\\ &&&&\displaystyle\frac{1}{2}y&&&&&&\displaystyle=&&\displaystyle 3/2&\\ &&&&&&&&\displaystyle\;-z&&\displaystyle\;=&&\displaystyle 1&\end{aligned}
  16. L 2 + 1 2 L 3 L 2 L_{2}+\frac{1}{2}L_{3}\rightarrow L_{2}
  17. L 1 - L 3 L 1 L_{1}-L_{3}\rightarrow L_{1}
  18. [ 2 1 0 7 0 1 / 2 0 3 / 2 0 0 - 1 1 ] \left[\begin{array}[]{ccc|c}2&1&0&7\\ 0&1/2&0&3/2\\ 0&0&-1&1\end{array}\right]
  19. 2 x + y = 7 y = 3 z = - 1 \begin{aligned}\displaystyle 2x&&\displaystyle\;+&&\displaystyle y&&&&&&% \displaystyle=&&\displaystyle 7&\\ &&&&\displaystyle y&&&&&&\displaystyle=&&\displaystyle 3&\\ &&&&&&&&\displaystyle\;z&&\displaystyle\;=&&\displaystyle-1&\end{aligned}
  20. 2 L 2 L 2 2L_{2}\rightarrow L_{2}
  21. - L 3 L 3 -L_{3}\rightarrow L_{3}
  22. [ 2 1 0 7 0 1 0 3 0 0 1 - 1 ] \left[\begin{array}[]{ccc|c}2&1&0&7\\ 0&1&0&3\\ 0&0&1&-1\end{array}\right]
  23. x = 2 y = 3 z = - 1 \begin{aligned}\displaystyle x&&&&&&&&&&\displaystyle=&&\displaystyle 2&\\ &&&&\displaystyle y&&&&&&\displaystyle=&&\displaystyle 3&\\ &&&&&&&&\displaystyle\;z&&\displaystyle\;=&&\displaystyle-1&\end{aligned}
  24. L 1 - L 2 L 1 L_{1}-L_{2}\rightarrow L_{1}
  25. 1 2 L 1 L 1 \frac{1}{2}L_{1}\rightarrow L_{1}
  26. [ 1 0 0 2 0 1 0 3 0 0 1 - 1 ] \left[\begin{array}[]{ccc|c}1&0&0&2\\ 0&1&0&3\\ 0&0&1&-1\end{array}\right]
  27. L 2 L_{2}
  28. 3 2 L 1 \begin{matrix}\frac{3}{2}\end{matrix}L_{1}
  29. L 2 L_{2}
  30. L 3 L_{3}
  31. L 1 L_{1}
  32. L 3 L_{3}
  33. L 2 + 3 2 L 1 L 2 L_{2}+\frac{3}{2}L_{1}\rightarrow L_{2}
  34. L 3 + L 1 L 3 . L_{3}+L_{1}\rightarrow L_{3}.
  35. A = [ 2 - 1 0 - 1 2 - 1 0 - 1 2 ] . A=\begin{bmatrix}2&-1&0\\ -1&2&-1\\ 0&-1&2\end{bmatrix}.
  36. [ A | I ] = [ 2 - 1 0 1 0 0 - 1 2 - 1 0 1 0 0 - 1 2 0 0 1 ] . [A|I]=\left[\begin{array}[]{rrr|rrr}2&-1&0&1&0&0\\ -1&2&-1&0&1&0\\ 0&-1&2&0&0&1\end{array}\right].
  37. [ I | B ] = [ 1 0 0 3 4 1 2 1 4 0 1 0 1 2 1 1 2 0 0 1 1 4 1 2 3 4 ] . [I|B]=\left[\begin{array}[]{rrr|rrr}1&0&0&\frac{3}{4}&\frac{1}{2}&\frac{1}{4}% \\ 0&1&0&\frac{1}{2}&1&\frac{1}{2}\\ 0&0&1&\frac{1}{4}&\frac{1}{2}&\frac{3}{4}\end{array}\right].
  38. m × n m\times n
  39. A A
  40. 6 × 9 6\times 9
  41. T = [ a * * * * * * * * 0 0 b * * * * * * 0 0 0 c * * * * * 0 0 0 0 0 0 d * * 0 0 0 0 0 0 0 0 e 0 0 0 0 0 0 0 0 0 ] T=\begin{bmatrix}a&*&*&*&*&*&*&*&*\\ 0&0&b&*&*&*&*&*&*\\ 0&0&0&c&*&*&*&*&*\\ 0&0&0&0&0&0&d&*&*\\ 0&0&0&0&0&0&0&0&e\\ 0&0&0&0&0&0&0&0&0\end{bmatrix}
  42. T T
  43. A A
  44. A A
  45. T T
  46. A A
  47. A A
  48. T T
  49. A A
  50. n 3 n^{3}
  51. m m
  52. n n
  53. A A
  54. m m
  55. m m
  56. S S
  57. T T
  58. S S
  59. T T
  60. A A
  61. A [ i , j ] A[i,j]
  62. i i
  63. j j
  64. A A
  65. A A
  66. T T

Gaussian_integer.html

  1. 𝐙 i i \mathbf{Z}ii
  2. 𝐙 [ i ] = { a + b i a , b 𝐙 } , where i 2 = - 1. \mathbf{Z}[i]=\{a+bi\mid a,b\in\mathbf{Z}\},\qquad\,\text{ where }i^{2}=-1.
  3. 2 2
  4. N ( a + b i ) = a 2 + b 2 = ( a + b i ) ( a + b i ) ¯ = ( a + b i ) ( a - b i ) , N(a+bi)=a^{2}+b^{2}=(a+bi)\overline{(a+bi)}=(a+bi)(a-bi),
  5. ¯ \overline{⋅}
  6. N ( z w ) = N ( z ) N ( w ) . N(zw)=N(z)N(w).
  7. 𝐙 i i \mathbf{Z}ii
  8. 1 1
  9. x 𝐙 i i x∈\mathbf{Z}ii
  10. ± x , ± i x ±x,±ix
  11. x x
  12. 𝐙 i i \mathbf{Z}ii
  13. 𝐙 i i \mathbf{Z}ii
  14. 3 3
  15. 4 4
  16. 𝐙 \mathbf{Z}
  17. a + b i a+bi
  18. a , b a,b
  19. 4 n + 3 4n+3
  20. n n
  21. ( 4 n + 3 ) −(4n+3)
  22. 4 n + 3 4n+3
  23. ± 1 , ± i ±1,±i
  24. 4 n + 3 4n+3
  25. p p
  26. p p
  27. p p
  28. p p
  29. p p
  30. 5 = ( 2 + i ) · ( 2 i ) 5=(2+i)·(2−i)
  31. 13 = ( 3 + 2 i ) · ( 3 2 i ) 13=(3+2i)·(3−2i)
  32. p = 2 p=2
  33. 𝐙 \mathbf{Z}
  34. 𝐐 ( i ) \mathbf{Q}(i)
  35. 2 2 \tfrac{\sqrt{2}}{2}
  36. 2 2 N ( z ) \frac{\sqrt{2}}{2}\sqrt{N(z)}
  37. z z
  38. z z
  39. 𝐙 i i \mathbf{Z}ii
  40. v ( z ) = N ( z ) . v(z)=N(z).
  41. x < s u p > 2 q ( m o d p ) x<sup>2≡q(modp)

Gaussian_quadrature.html

  1. 2 n 1 2n−1
  2. i = 1 , , n i=1,...,n
  3. - 1 1 f ( x ) d x = i = 1 n w i f ( x i ) . \int_{-1}^{1}f(x)\,dx=\sum_{i=1}^{n}w_{i}f(x_{i}).
  4. 1 , 11 −1,11
  5. f ( x ) = ω ( x ) g ( x ) f(x)=\omega(x)g(x)\,
  6. g ( x ) g(x)
  7. ω ( x ) ω(x)
  8. w i w_{i}^{\prime}
  9. x i x_{i}^{\prime}
  10. ω ( x ) ω(x)
  11. - 1 1 f ( x ) d x = - 1 1 ω ( x ) g ( x ) d x i = 1 n w i g ( x i ) . \int_{-1}^{1}f(x)\,dx=\int_{-1}^{1}\omega(x)g(x)\,dx\approx\sum_{i=1}^{n}w_{i}% ^{\prime}g(x_{i}^{\prime}).
  12. ω ( x ) = 1 / 1 - x 2 \omega(x)=1/\sqrt{1-x^{2}}\,
  13. ω ( x ) = e - x 2 \omega(x)=e^{-x^{2}}
  14. ω ( x ) = 1 \omega(x)=1
  15. n n
  16. i i
  17. i i
  18. w i = 2 ( 1 - x i 2 ) [ P n ( x i ) ] 2 . w_{i}=\frac{2}{\left(1-x_{i}^{2}\right)[P^{\prime}_{n}(x_{i})]^{2}}.
  19. ± 1 3 \pm\sqrt{\tfrac{1}{3}}
  20. 8 9 \tfrac{8}{9}
  21. ± 3 5 \pm\sqrt{\tfrac{3}{5}}
  22. 5 9 \tfrac{5}{9}
  23. ± 3 7 - 2 7 6 5 \pm\sqrt{\tfrac{3}{7}-\tfrac{2}{7}\sqrt{\tfrac{6}{5}}}
  24. 18 + 30 36 \tfrac{18+\sqrt{30}}{36}
  25. ± 3 7 + 2 7 6 5 \pm\sqrt{\tfrac{3}{7}+\tfrac{2}{7}\sqrt{\tfrac{6}{5}}}
  26. 18 - 30 36 \tfrac{18-\sqrt{30}}{36}
  27. 128 225 \tfrac{128}{225}
  28. ± 1 3 5 - 2 10 7 \pm\tfrac{1}{3}\sqrt{5-2\sqrt{\tfrac{10}{7}}}
  29. 322 + 13 70 900 \tfrac{322+13\sqrt{70}}{900}
  30. ± 1 3 5 + 2 10 7 \pm\tfrac{1}{3}\sqrt{5+2\sqrt{\tfrac{10}{7}}}
  31. 322 - 13 70 900 \tfrac{322-13\sqrt{70}}{900}
  32. a a , b aa,b
  33. 1 , 11 −1,11
  34. a b f ( x ) d x = b - a 2 - 1 1 f ( b - a 2 x + a + b 2 ) d x . \int_{a}^{b}f(x)\,dx=\frac{b-a}{2}\int_{-1}^{1}f\left(\frac{b-a}{2}x+\frac{a+b% }{2}\right)\,dx.
  35. a b f ( x ) d x = b - a 2 i = 1 n w i f ( b - a 2 x i + a + b 2 ) . \int_{a}^{b}f(x)\,dx=\frac{b-a}{2}\sum_{i=1}^{n}w_{i}f\left(\frac{b-a}{2}x_{i}% +\frac{a+b}{2}\right).
  36. 1 , 11 −1,11
  37. a b ω ( x ) f ( x ) d x \int_{a}^{b}\omega(x)\,f(x)\,dx
  38. ω ( x ) ω(x)
  39. 1 1
  40. ( 1 - x ) α ( 1 + x ) β , α , β > - 1 (1-x)^{\alpha}(1+x)^{\beta},\quad\alpha,\beta>-1
  41. β = 0 β=0
  42. 1 1 - x 2 \frac{1}{\sqrt{1-x^{2}}}
  43. 1 - x 2 \sqrt{1-x^{2}}
  44. e - x e^{-x}\,
  45. x α e - x , α > - 1 x^{\alpha}e^{-x},\quad\alpha>-1
  46. e - x 2 e^{-x^{2}}
  47. a b ω ( x ) x k p n ( x ) d x = 0 , for all k = 0 , 1 , , n - 1. \int_{a}^{b}\omega(x)\,x^{k}p_{n}(x)\,dx=0,\quad\,\text{for all }k=0,1,\ldots,% n-1.
  48. h ( x ) h(x)
  49. ω ( x ) ω(x)
  50. h ( x ) h(x)
  51. p n ( x ) p_{n}(x)
  52. q ( x ) q(x)
  53. r ( x ) r(x)
  54. p n ( x ) p_{n}(x)
  55. p n ( x ) p_{n}(x)
  56. a b ω ( x ) h ( x ) d x = a b ω ( x ) r ( x ) d x . \int_{a}^{b}\omega(x)\,h(x)\,dx=\int_{a}^{b}\omega(x)\,r(x)\,dx.
  57. i = 1 n w i h ( x i ) = i = 1 n w i r ( x i ) \sum_{i=1}^{n}w_{i}h(x_{i})=\sum_{i=1}^{n}w_{i}r(x_{i})
  58. h ( x ) h(x)
  59. r ( x ) r(x)
  60. w i = a n a n - 1 a b ω ( x ) p n - 1 ( x ) 2 d x p n ( x i ) p n - 1 ( x i ) w_{i}=\frac{a_{n}}{a_{n-1}}\frac{\int_{a}^{b}\omega(x)p_{n-1}(x)^{2}dx}{p^{% \prime}_{n}(x_{i})p_{n-1}(x_{i})}
  61. a k a_{k}
  62. x k x^{k}
  63. p k ( x ) p_{k}(x)
  64. r ( x ) r(x)
  65. r ( x i ) r(x_{i})
  66. r ( x ) = i = 1 n r ( x i ) 1 j n j i x - x j x i - x j r(x)=\sum_{i=1}^{n}r(x_{i})\prod_{\begin{smallmatrix}1\leq j\leq n\\ j\neq i\end{smallmatrix}}\frac{x-x_{j}}{x_{i}-x_{j}}
  67. r ( x ) r(x)
  68. n n
  69. n n
  70. ω ( x ) ω(x)
  71. a a
  72. b b
  73. a b ω ( x ) r ( x ) d x = i = 1 n r ( x i ) a b ω ( x ) 1 j n j i x - x j x i - x j d x \int_{a}^{b}\omega(x)r(x)dx=\sum_{i=1}^{n}r(x_{i})\int_{a}^{b}\omega(x)\prod_{% \begin{smallmatrix}1\leq j\leq n\\ j\neq i\end{smallmatrix}}\frac{x-x_{j}}{x_{i}-x_{j}}dx
  74. w i = a b ω ( x ) 1 j n j i x - x j x i - x j d x w_{i}=\int_{a}^{b}\omega(x)\prod_{\begin{smallmatrix}1\leq j\leq n\\ j\neq i\end{smallmatrix}}\frac{x-x_{j}}{x_{i}-x_{j}}dx
  75. w i w_{i}
  76. p n ( x ) p_{n}(x)
  77. p n + 1 ( x ) p_{n+1}(x)
  78. 1 j n j i ( x - x j ) = 1 j n ( x - x j ) x - x i = p n ( x ) a n ( x - x i ) \prod_{\begin{smallmatrix}1\leq j\leq n\\ j\neq i\end{smallmatrix}}\left(x-x_{j}\right)=\frac{\prod_{1\leq j\leq n}\left% (x-x_{j}\right)}{x-x_{i}}=\frac{p_{n}(x)}{a_{n}\left(x-x_{i}\right)}
  79. a n a_{n}
  80. x n x^{n}
  81. p n ( x ) p_{n}(x)
  82. x i x_{i}
  83. 1 j n j i ( x i - x j ) = p n ( x i ) a n \prod_{\begin{smallmatrix}1\leq j\leq n\\ j\neq i\end{smallmatrix}}\left(x_{i}-x_{j}\right)=\frac{p^{\prime}_{n}(x_{i})}% {a_{n}}
  84. w i = 1 p n ( x i ) a b ω ( x ) p n ( x ) x - x i d x w_{i}=\frac{1}{p^{\prime}_{n}(x_{i})}\int_{a}^{b}\omega(x)\frac{p_{n}(x)}{x-x_% {i}}dx
  85. 1 x - x i = 1 - ( x x i ) k x - x i + ( x x i ) k 1 x - x i \frac{1}{x-x_{i}}=\frac{1-\left(\frac{x}{x_{i}}\right)^{k}}{x-x_{i}}+\left(% \frac{x}{x_{i}}\right)^{k}\frac{1}{x-x_{i}}
  86. a b ω ( x ) x k p n ( x ) x - x i d x = x i k a b ω ( x ) p n ( x ) x - x i d x \int_{a}^{b}\omega(x)\frac{x^{k}p_{n}(x)}{x-x_{i}}dx=x_{i}^{k}\int_{a}^{b}% \omega(x)\frac{p_{n}(x)}{x-x_{i}}dx
  87. k n k\leq n
  88. 1 - ( x x i ) k x - x i \frac{1-\left(\frac{x}{x_{i}}\right)^{k}}{x-x_{i}}
  89. p n ( x ) p_{n}(x)
  90. q ( x ) q(x)
  91. a b ω ( x ) p n ( x ) x - x i d x = 1 q ( x i ) a b ω ( x ) q ( x ) p n ( x ) x - x i d x \int_{a}^{b}\omega(x)\frac{p_{n}(x)}{x-x_{i}}dx=\frac{1}{q(x_{i})}\int_{a}^{b}% \omega(x)\frac{q(x)p_{n}(x)}{x-x_{i}}dx
  92. q ( x ) = p n - 1 ( x ) q(x)=p_{n-1}(x)
  93. p n ( x ) x - x i \frac{p_{n}(x)}{x-x_{i}}
  94. p n ( x ) x - x i = a n x n - 1 + s ( x ) \frac{p_{n}(x)}{x-x_{i}}=a_{n}x^{n-1}+s(x)
  95. s ( x ) s(x)
  96. n - 2 n-2
  97. s ( x ) s(x)
  98. p n - 1 ( x ) p_{n-1}(x)
  99. a b ω ( x ) p n ( x ) x - x i d x = a n p n - 1 ( x i ) a b ω ( x ) p n - 1 ( x ) x n - 1 d x \int_{a}^{b}\omega(x)\frac{p_{n}(x)}{x-x_{i}}dx=\frac{a_{n}}{p_{n-1}(x_{i})}% \int_{a}^{b}\omega(x)p_{n-1}(x)x^{n-1}dx
  100. x n - 1 = ( x n - 1 - p n - 1 ( x ) a n - 1 ) + p n - 1 ( x ) a n - 1 x^{n-1}=\left(x^{n-1}-\frac{p_{n-1}(x)}{a_{n-1}}\right)+\frac{p_{n-1}(x)}{a_{n% -1}}
  101. n - 2 n-2
  102. p n - 1 ( x ) p_{n-1}(x)
  103. a b ω ( x ) p n ( x ) x - x i d x = a n a n - 1 p n - 1 ( x i ) a b ω ( x ) p n - 1 ( x ) 2 d x \int_{a}^{b}\omega(x)\frac{p_{n}(x)}{x-x_{i}}dx=\frac{a_{n}}{a_{n-1}p_{n-1}(x_% {i})}\int_{a}^{b}\omega(x)p_{n-1}(x)^{2}dx
  104. p n ( x i ) p^{\prime}_{n}(x_{i})
  105. f ( x ) = 1 j n j i ( x - x j ) 2 f(x)=\prod_{\begin{smallmatrix}1\leq j\leq n\\ j\neq i\end{smallmatrix}}(x-x_{j})^{2}
  106. p n ( x ) p_{n}(x)
  107. p n ( x ) p_{n}(x)
  108. f ( x j ) = 0 f(x_{j})=0
  109. a b ω ( x ) f ( x ) d x = j = 1 N w j f ( x j ) = w i f ( x i ) . \int_{a}^{b}\omega(x)f(x)dx=\sum_{j=1}^{N}w_{j}f(x_{j})=w_{i}f(x_{i}).
  110. ω ( x ) \omega(x)
  111. w i > 0 w_{i}>0
  112. n n
  113. p r p_{r}
  114. ( p r , p s ) = 0 (p_{r},p_{s})=0
  115. r s r\neq s
  116. ( , ) (,)
  117. ( p r ) = r (p_{r})=r
  118. p r + 1 ( x ) = ( x - a r , r ) p r ( x ) - a r , r - 1 p r - 1 ( x ) - a r , 0 p 0 ( x ) p_{r+1}(x)=(x-a_{r,r})p_{r}(x)-a_{r,r-1}p_{r-1}(x)\ldots-a_{r,0}p_{0}(x)
  119. r = 0 , 1 , , n - 1 r=0,1,\ldots,n-1
  120. n n
  121. a r , s = ( x p r , p s ) / ( p s , p s ) a_{r,s}=(xp_{r},p_{s})/(p_{s},p_{s})
  122. p 0 ( x ) = 1 p_{0}(x)=1
  123. p 0 p_{0}
  124. p r p_{r}
  125. r = s = 0 r=s=0
  126. ( p 1 , p 0 ) = ( ( x - a 0 , 0 p 0 , p 0 ) = ( x p 0 , p 0 ) - a 0 , 0 ( p 0 , p 0 ) = ( x p 0 , p 0 ) - ( x p 0 , p 0 ) = 0. (p_{1},p_{0})=((x-a_{0,0}p_{0},p_{0})=(xp_{0},p_{0})-a_{0,0}(p_{0},p_{0})=(xp_% {0},p_{0})-(xp_{0},p_{0})=0.
  127. p 0 , p 1 , , p r p_{0},p_{1},\ldots,p_{r}
  128. p r + 1 p_{r+1}
  129. ( p r + 1 , p s ) = ( x p r , p s ) - a r , r ( p r , p s ) - a r , r - 1 ( p r - 1 , p s ) - a r , 0 ( p 0 , p s ) (p_{r+1},p_{s})=(xp_{r},p_{s})-a_{r,r}(p_{r},p_{s})-a_{r,r-1}(p_{r-1},p_{s})% \ldots-a_{r,0}(p_{0},p_{s})
  130. p s p_{s}
  131. ( p r + 1 , p s ) = ( x p r , p s ) - a r , s ( p s , p s ) = ( x p r , p s ) - ( x p r , p s ) = 0. (p_{r+1},p_{s})=(xp_{r},p_{s})-a_{r,s}(p_{s},p_{s})=(xp_{r},p_{s})-(xp_{r},p_{% s})=0.
  132. ( x f , g ) = ( f , x g ) (xf,g)=(f,xg)
  133. s r - 1 , x p s s\leq r-1,xp_{s}
  134. r 1 r−1
  135. p r p_{r}
  136. r 1 r−1
  137. ( x p r , p s ) = ( p r , x p s ) = 0 (xp_{r},p_{s})=(p_{r},xp_{s})=0
  138. a r , s = 0 a_{r,s}=0
  139. p r + 1 ( x ) = ( x - a r , r ) p r ( x ) - a r , r - 1 p r - 1 ( x ) p_{r+1}(x)=(x-a_{r,r})p_{r}(x)-a_{r,r-1}p_{r-1}(x)
  140. p r + 1 ( x ) = ( x - a r ) p r ( x ) - b r p r - 1 ( x ) p_{r+1}(x)=(x-a_{r})p_{r}(x)-b_{r}p_{r-1}(x)
  141. p - 1 ( x ) 0 p_{-1}(x)\equiv 0
  142. a r := ( x p r , p r ) ( p r , p r ) , b r := ( x p r , p r - 1 ) ( p r - 1 , p r - 1 ) = ( p r , p r ) ( p r - 1 , p r - 1 ) a_{r}:=\frac{(xp_{r},p_{r})}{(p_{r},p_{r})},\qquad b_{r}:=\frac{(xp_{r},p_{r-1% })}{(p_{r-1},p_{r-1})}=\frac{(p_{r},p_{r})}{(p_{r-1},p_{r-1})}
  143. ( x p r , p r - 1 ) = ( p r , x p r - 1 ) = ( p r , p r ) (xp_{r},p_{r-1})=(p_{r},xp_{r-1})=(p_{r},p_{r})
  144. x p r - 1 xp_{r-1}
  145. p r p_{r}
  146. r r
  147. J P ~ = x P ~ - p n ( x ) × 𝐞 n J\tilde{P}=x\tilde{P}-p_{n}(x)\times\mathbf{e}_{n}
  148. P ~ = [ p 0 ( x ) , p 1 ( x ) , , p n - 1 ( x ) ] T \tilde{P}=[p_{0}(x),p_{1}(x),...,p_{n-1}(x)]^{T}
  149. 𝐞 n \mathbf{e}_{n}
  150. n n
  151. 𝐞 n = [ 0 , , 0 , 1 ] T \mathbf{e}_{n}=[0,...,0,1]^{T}
  152. J J
  153. 𝐉 = ( a 0 1 0 b 1 a 1 1 0 0 b 2 a 2 1 0 0 0 0 b n - 2 a n - 2 1 0 b n - 1 a n - 1 ) \mathbf{J}=\begin{pmatrix}a_{0}&1&0&\ldots&\ldots&\ldots\\ b_{1}&a_{1}&1&0&\ldots&\ldots\\ 0&b_{2}&a_{2}&1&0&\ldots\\ 0&\ldots&\ldots&\ldots&\ldots&0\\ \ldots&\ldots&0&b_{n-2}&a_{n-2}&1\\ \ldots&\ldots&\ldots&0&b_{n-1}&a_{n-1}\end{pmatrix}
  154. x j x_{j}
  155. n n
  156. 𝒥 \mathcal{J}
  157. 𝒥 i , i = J i , i = a i - 1 i = 1 , , n 𝒥 i - 1 , i = 𝒥 i , i - 1 = J i , i - 1 J i - 1 , i = b i - 1 i = 2 , , n . \begin{aligned}\displaystyle\mathcal{J}_{i,i}&\displaystyle=J_{i,i}=a_{i-1}&&% \displaystyle i=1,\ldots,n\\ \displaystyle\mathcal{J}_{i-1,i}=\mathcal{J}_{i,i-1}&\displaystyle=\sqrt{J_{i,% i-1}J_{i-1,i}}=\sqrt{b_{i-1}}&&\displaystyle i=2,\ldots,n.\end{aligned}
  158. 𝐉 \mathbf{J}
  159. 𝒥 \mathcal{J}
  160. ϕ ( j ) \phi^{(j)}
  161. w j = μ 0 ( ϕ 1 ( j ) ) 2 w_{j}=\mu_{0}\left(\phi_{1}^{(j)}\right)^{2}
  162. μ 0 \mu_{0}
  163. μ 0 = a b ω ( x ) d x . \mu_{0}=\int_{a}^{b}\omega(x)dx.
  164. 2 n 2n
  165. a b ω ( x ) f ( x ) d x - i = 1 n w i f ( x i ) = f ( 2 n ) ( ξ ) ( 2 n ) ! ( p n , p n ) \int_{a}^{b}\omega(x)\,f(x)\,dx-\sum_{i=1}^{n}w_{i}\,f(x_{i})=\frac{f^{(2n)}(% \xi)}{(2n)!}\,(p_{n},p_{n})
  166. ξ ξ
  167. ( a , b ) (a,b)
  168. n n
  169. ( f , g ) = a b ω ( x ) f ( x ) g ( x ) d x . (f,g)=\int_{a}^{b}\omega(x)f(x)g(x)\,dx.
  170. ω ( x ) = 1 ω(x)=1
  171. ( b - a ) 2 n + 1 ( n ! ) 4 ( 2 n + 1 ) [ ( 2 n ) ! ] 3 f ( 2 n ) ( ξ ) , a < ξ < b . \frac{(b-a)^{2n+1}(n!)^{4}}{(2n+1)[(2n)!]^{3}}f^{(2n)}(\xi),\qquad a<\xi<b.
  172. n n
  173. 2 n 1 2n−1
  174. a a , b aa,b
  175. n + 1 n+1
  176. n n
  177. 3 n + 1 3n+1
  178. 1 , 11 −1,11
  179. - 1 1 f ( x ) d x = 2 n ( n - 1 ) [ f ( 1 ) + f ( - 1 ) ] + i = 2 n - 1 w i f ( x i ) + R n . \int_{-1}^{1}{f(x)\,dx}=\frac{2}{n(n-1)}[f(1)+f(-1)]+\sum_{i=2}^{n-1}{w_{i}f(x% _{i})}+R_{n}.
  180. ( i - 1 ) (i-1)
  181. P n - 1 ( x ) P^{\prime}_{n-1}(x)
  182. w i = 2 n ( n - 1 ) [ P n - 1 ( x i ) ] 2 , x i ± 1. w_{i}=\frac{2}{n(n-1)[P_{n-1}(x_{i})]^{2}},\qquad x_{i}\neq\pm 1.
  183. R n = - n ( n - 1 ) 3 2 2 n - 1 [ ( n - 2 ) ! ] 4 ( 2 n - 1 ) [ ( 2 n - 2 ) ! ] 3 f ( 2 n - 2 ) ( ξ ) , - 1 < ξ < 1. R_{n}=\frac{-n(n-1)^{3}2^{2n-1}[(n-2)!]^{4}}{(2n-1)[(2n-2)!]^{3}}f^{(2n-2)}(% \xi),\qquad-1<\xi<1.
  184. 3 3
  185. 0
  186. 4 3 \frac{4}{3}
  187. ± 1 \pm 1
  188. 1 3 \frac{1}{3}
  189. 4 4
  190. ± 1 5 \pm\sqrt{\frac{1}{5}}
  191. 5 6 \frac{5}{6}
  192. ± 1 \pm 1
  193. 1 6 \frac{1}{6}
  194. 5 5
  195. 0
  196. 32 45 \frac{32}{45}
  197. ± 3 7 \pm\sqrt{\frac{3}{7}}
  198. 49 90 \frac{49}{90}
  199. ± 1 \pm 1
  200. 1 10 \frac{1}{10}