wpmath0000001_2

Bayes'_theorem.html

  1. P ( A | B ) = P ( B | A ) P ( A ) P ( B ) , P(A|B)=\frac{P(B|A)\,P(A)}{P(B)},
  2. 0.5 % ÷ 0.2 % * 1 % = 2.5 % 0.5\%\div 0.2\%*1\%=2.5\%
  3. P ( User + ) = P ( + User ) P ( User ) P ( + User ) P ( User ) + P ( + Non-user ) P ( Non-user ) = 0.99 × 0.005 0.99 × 0.005 + 0.01 × 0.995 33.2 % \begin{aligned}\displaystyle P(\,\text{User}\mid\,\text{+})&\displaystyle=% \frac{P(\,\text{+}\mid\,\text{User})P(\,\text{User})}{P(\,\text{+}\mid\,\text{% User})P(\,\text{User})+P(\,\text{+}\mid\,\text{Non-user})P(\,\text{Non-user})}% \\ &\displaystyle=\frac{0.99\times 0.005}{0.99\times 0.005+0.01\times 0.995}\\ &\displaystyle\approx 33.2\%\end{aligned}
  4. P ( Rare Pattern ) \displaystyle P(\,\text{Rare}\mid\,\text{Pattern})
  5. P ( A B ) = P ( B A ) P ( A ) P ( B ) P(A\mid B)=\frac{P(B\mid A)\,P(A)}{P(B)}\cdot\,
  6. P ( A B ) P ( A ) P ( B A ) P(A\mid B)\propto P(A)\cdot P(B\mid A)
  7. P ( A B ) = c P ( A ) P ( B A ) P(A\mid B)=c\cdot P(A)\cdot P(B\mid A)
  8. P ( ¬ A B ) = c P ( ¬ A ) P ( B ¬ A ) P(\neg A\mid B)=c\cdot P(\neg A)\cdot P(B\mid\neg A)\cdot
  9. c = 1 P ( A ) P ( B A ) + P ( ¬ A ) P ( B ¬ A ) . c=\frac{1}{P(A)\cdot P(B\mid A)+P(\neg A)\cdot P(B\mid\neg A)}.
  10. P ( A B ) = P ( B A ) P ( A ) P ( B A ) P ( A ) + P ( B ¬ A ) P ( ¬ A ) P(A\mid B)=\frac{P(B\mid A)\,P(A)}{P(B\mid A)P(A)+P(B\mid\neg A)P(\neg A)}\cdot
  11. P ( B ) = j P ( B A j ) P ( A j ) , P(B)={\sum_{j}P(B\mid A_{j})P(A_{j})},
  12. P ( A i B ) = P ( B A i ) P ( A i ) j P ( B A j ) P ( A j ) \Rightarrow P(A_{i}\mid B)=\frac{P(B\mid A_{i})\,P(A_{i})}{\sum\limits_{j}P(B% \mid A_{j})\,P(A_{j})}\cdot
  13. P ( A B ) = P ( B A ) P ( A ) P ( B A ) P ( A ) + P ( B ¬ A ) P ( ¬ A ) P(A\mid B)=\frac{P(B\mid A)\,P(A)}{P(B\mid A)P(A)+P(B\mid\neg A)P(\neg A)}\cdot
  14. f X ( x Y = y ) = P ( Y = y X = x ) f X ( x ) P ( Y = y ) . f_{X}(x\mid Y=y)=\frac{P(Y=y\mid X=x)\,f_{X}(x)}{P(Y=y)}.
  15. P ( X = x Y = y ) = f Y ( y X = x ) P ( X = x ) f Y ( y ) . P(X=x\mid Y=y)=\frac{f_{Y}(y\mid X=x)\,P(X=x)}{f_{Y}(y)}.
  16. f X ( x Y = y ) = f Y ( y X = x ) f X ( x ) f Y ( y ) . f_{X}(x\mid Y=y)=\frac{f_{Y}(y\mid X=x)\,f_{X}(x)}{f_{Y}(y)}.
  17. f Y ( y ) = - f Y ( y X = ξ ) f X ( ξ ) d ξ . f_{Y}(y)=\int_{-\infty}^{\infty}f_{Y}(y\mid X=\xi)\,f_{X}(\xi)\,d\xi.
  18. O ( A 1 : A 2 B ) = O ( A 1 : A 2 ) Λ ( A 1 : A 2 B ) O(A_{1}:A_{2}\mid B)=O(A_{1}:A_{2})\cdot\Lambda(A_{1}:A_{2}\mid B)
  19. Λ ( A 1 : A 2 B ) = P ( B A 1 ) P ( B A 2 ) \Lambda(A_{1}:A_{2}\mid B)=\frac{P(B\mid A_{1})}{P(B\mid A_{2})}
  20. O ( A 1 : A 2 ) = P ( A 1 ) P ( A 2 ) , O(A_{1}:A_{2})=\frac{P(A_{1})}{P(A_{2})},
  21. O ( A 1 : A 2 B ) = P ( A 1 B ) P ( A 2 B ) , O(A_{1}:A_{2}\mid B)=\frac{P(A_{1}\mid B)}{P(A_{2}\mid B)},
  22. P ( A B ) = P ( A B ) P ( B ) , if P ( B ) 0 , P(A\mid B)=\frac{P(A\cap B)}{P(B)},\,\text{ if }P(B)\neq 0,\!
  23. P ( B A ) = P ( A B ) P ( A ) , if P ( A ) 0 , P(B\mid A)=\frac{P(A\cap B)}{P(A)},\,\text{ if }P(A)\neq 0,\!
  24. P ( A B ) = P ( A B ) P ( B ) = P ( B A ) P ( A ) , \Rightarrow P(A\cap B)=P(A\mid B)\,P(B)=P(B\mid A)\,P(A),\!
  25. P ( A B ) = P ( B A ) P ( A ) P ( B ) , if P ( B ) 0. \Rightarrow P(A\mid B)=\frac{P(B\mid A)\,P(A)}{P(B)},\,\text{ if }P(B)\neq 0.
  26. f X ( x Y = y ) = f X , Y ( x , y ) f Y ( y ) f_{X}(x\mid Y=y)=\frac{f_{X,Y}(x,y)}{f_{Y}(y)}
  27. f Y ( y X = x ) = f X , Y ( x , y ) f X ( x ) f_{Y}(y\mid X=x)=\frac{f_{X,Y}(x,y)}{f_{X}(x)}
  28. f X ( x Y = y ) = f Y ( y X = x ) f X ( x ) f Y ( y ) . \Rightarrow f_{X}(x\mid Y=y)=\frac{f_{Y}(y\mid X=x)\,f_{X}(x)}{f_{Y}(y)}.

Bayesian_inference.html

  1. P ( H E ) = P ( E H ) P ( H ) P ( E ) P(H\mid E)=\frac{P(E\mid H)\cdot P(H)}{P(E)}
  2. \textstyle\mid
  3. H \textstyle H
  4. E \textstyle E
  5. P ( H ) \textstyle P(H)
  6. H \textstyle H
  7. E \textstyle E
  8. P ( H E ) \textstyle P(H\mid E)
  9. H \textstyle H
  10. E \textstyle E
  11. E \textstyle E
  12. P ( E H ) \textstyle P(E\mid H)
  13. E \textstyle E
  14. H \textstyle H
  15. H \textstyle H
  16. E \textstyle E
  17. P ( H E ) \textstyle P(H\mid E)
  18. H \textstyle H
  19. E \textstyle E
  20. P ( E ) \textstyle P(E)
  21. H \textstyle H
  22. H \textstyle H
  23. P ( H ) \textstyle P(H)
  24. P ( E H ) \textstyle P(E\mid H)
  25. P ( H E ) \textstyle P(H\mid E)
  26. P ( H E ) = P ( E H ) P ( E ) P ( H ) P(H\mid E)=\frac{P(E\mid H)}{P(E)}\cdot P(H)
  27. P ( E H ) P ( E ) \textstyle\frac{P(E\mid H)}{P(E)}
  28. E E
  29. H H
  30. H 1 \textstyle H_{1}
  31. H 2 \textstyle H_{2}
  32. H 3 \textstyle H_{3}
  33. H 2 \textstyle H_{2}
  34. H 1 \textstyle H_{1}
  35. H 3 \textstyle H_{3}
  36. H 3 \textstyle H_{3}
  37. x x
  38. θ \theta
  39. x p ( x θ ) x\sim p(x\mid\theta)
  40. α \alpha
  41. θ p ( θ α ) \theta\sim p(\theta\mid\alpha)
  42. 𝐗 \mathbf{X}
  43. n n
  44. x 1 , , x n x_{1},\ldots,x_{n}
  45. x ~ \tilde{x}
  46. p ( θ α ) p(\theta\mid\alpha)
  47. p ( 𝐗 θ ) p(\mathbf{X}\mid\theta)
  48. L ( θ 𝐗 ) = p ( 𝐗 θ ) \operatorname{L}(\theta\mid\mathbf{X})=p(\mathbf{X}\mid\theta)
  49. p ( 𝐗 α ) = θ p ( 𝐗 θ ) p ( θ α ) d θ p(\mathbf{X}\mid\alpha)=\int_{\theta}p(\mathbf{X}\mid\theta)p(\theta\mid\alpha% )\operatorname{d}\!\theta
  50. p ( θ 𝐗 , α ) = p ( 𝐗 θ ) p ( θ α ) p ( 𝐗 α ) p ( 𝐗 θ ) p ( θ α ) p(\theta\mid\mathbf{X},\alpha)=\frac{p(\mathbf{X}\mid\theta)p(\theta\mid\alpha% )}{p(\mathbf{X}\mid\alpha)}\propto p(\mathbf{X}\mid\theta)p(\theta\mid\alpha)
  51. p ( x ~ 𝐗 , α ) = θ p ( x ~ θ ) p ( θ 𝐗 , α ) d θ p(\tilde{x}\mid\mathbf{X},\alpha)=\int_{\theta}p(\tilde{x}\mid\theta)p(\theta% \mid\mathbf{X},\alpha)\operatorname{d}\!\theta
  52. p ( x ~ α ) = θ p ( x ~ θ ) p ( θ α ) d θ p(\tilde{x}\mid\alpha)=\int_{\theta}p(\tilde{x}\mid\theta)p(\theta\mid\alpha)% \operatorname{d}\!\theta
  53. Ω \Omega
  54. E n E_{n}
  55. Ω \Omega
  56. M m M_{m}
  57. P ( E n M m ) P(E_{n}\mid M_{m})
  58. P ( M m ) P(M_{m})
  59. M m M_{m}
  60. { P ( M m ) } \{P(M_{m})\}
  61. E { E n } \textstyle E\in\{E_{n}\}
  62. M { M m } M\in\{M_{m}\}
  63. P ( M ) P(M)
  64. P ( M E ) P(M\mid E)
  65. P ( M E ) = P ( E M ) m P ( E M m ) P ( M m ) P ( M ) P(M\mid E)=\frac{P(E\mid M)}{\sum_{m}{P(E\mid M_{m})P(M_{m})}}\cdot P(M)
  66. 𝐄 = { e 1 , , e n } \mathbf{E}=\{e_{1},\dots,e_{n}\}
  67. P ( M 𝐄 ) = P ( 𝐄 M ) m P ( 𝐄 M m ) P ( M m ) P ( M ) P(M\mid\mathbf{E})=\frac{P(\mathbf{E}\mid M)}{\sum_{m}{P(\mathbf{E}\mid M_{m})% P(M_{m})}}\cdot P(M)
  68. P ( 𝐄 M ) = k P ( e k M ) . P(\mathbf{E}\mid M)=\prod_{k}{P(e_{k}\mid M)}.
  69. θ \mathbf{\theta}
  70. θ \mathbf{\theta}
  71. p ( θ α ) p(\mathbf{\theta}\mid\mathbf{\alpha})
  72. α \mathbf{\alpha}
  73. 𝐄 = { e 1 , , e n } \mathbf{E}=\{e_{1},\dots,e_{n}\}
  74. e i e_{i}
  75. p ( e θ ) p(e\mid\mathbf{\theta})
  76. θ \mathbf{\theta}
  77. θ \mathbf{\theta}
  78. p ( θ 𝐄 , α ) \displaystyle p(\mathbf{\theta}\mid\mathbf{E},\mathbf{\alpha})
  79. p ( 𝐄 θ , α ) = k p ( e k θ ) p(\mathbf{E}\mid\mathbf{\theta},\mathbf{\alpha})=\prod_{k}p(e_{k}\mid\mathbf{% \theta})
  80. P ( E M ) P ( E ) > 1 P ( E M ) > P ( E ) \textstyle\frac{P(E\mid M)}{P(E)}>1\Rightarrow\textstyle P(E\mid M)>P(E)
  81. P ( E M ) P ( E ) = 1 P ( E M ) = P ( E ) \textstyle\frac{P(E\mid M)}{P(E)}=1\Rightarrow\textstyle P(E\mid M)=P(E)
  82. P ( M ) = 0 P(M)=0
  83. P ( M E ) = 0 P(M\mid E)=0
  84. P ( M ) = 1 P(M)=1
  85. P ( M | E ) = 1 P(M|E)=1
  86. M M
  87. M M
  88. 1 - P ( M ) = 0 1-P(M)=0
  89. 1 - P ( M E ) = 0 1-P(M\mid E)=0
  90. θ ~ = E [ θ ] = θ θ p ( θ 𝐗 , α ) d θ \tilde{\theta}=\operatorname{E}[\theta]=\int_{\theta}\theta\,p(\theta\mid% \mathbf{X},\alpha)\,d\theta
  91. { θ MAP } arg max θ p ( θ 𝐗 , α ) . \{\theta_{\,\text{MAP}}\}\subset\arg\max_{\theta}p(\theta\mid\mathbf{X},\alpha).
  92. x ~ \tilde{x}
  93. p ( x ~ | 𝐗 , α ) = θ p ( x ~ , θ 𝐗 , α ) d θ = θ p ( x ~ θ ) p ( θ 𝐗 , α ) d θ . p(\tilde{x}|\mathbf{X},\alpha)=\int_{\theta}p(\tilde{x},\theta\mid\mathbf{X},% \alpha)\,d\theta=\int_{\theta}p(\tilde{x}\mid\theta)p(\theta\mid\mathbf{X},% \alpha)\,d\theta.
  94. H 1 H_{1}
  95. H 2 H_{2}
  96. P ( H 1 ) = P ( H 2 ) P(H_{1})=P(H_{2})
  97. E E
  98. P ( E H 1 ) = 30 / 40 = 0.75 P(E\mid H_{1})=30/40=0.75
  99. P ( E H 2 ) = 20 / 40 = 0.5. P(E\mid H_{2})=20/40=0.5.
  100. P ( H 1 E ) \displaystyle P(H_{1}\mid E)
  101. P ( H 1 ) P(H_{1})
  102. P ( H 1 E ) P(H_{1}\mid E)
  103. C C
  104. { G D , G D ¯ , G ¯ D , G ¯ D ¯ } \{GD,G\bar{D},\bar{G}D,\bar{G}\bar{D}\}
  105. P ( E = G D C = c ) = ( 0.01 + 0.16 ( c - 11 ) ) ( 0.5 - 0.09 ( c - 11 ) ) P(E=GD\mid C=c)=(0.01+0.16(c-11))(0.5-0.09(c-11))
  106. P ( E = G D ¯ C = c ) = ( 0.01 + 0.16 ( c - 11 ) ) ( 0.5 + 0.09 ( c - 11 ) ) P(E=G\bar{D}\mid C=c)=(0.01+0.16(c-11))(0.5+0.09(c-11))
  107. P ( E = G ¯ D C = c ) = ( 0.99 - 0.16 ( c - 11 ) ) ( 0.5 - 0.09 ( c - 11 ) ) P(E=\bar{G}D\mid C=c)=(0.99-0.16(c-11))(0.5-0.09(c-11))
  108. P ( E = G ¯ D ¯ C = c ) = ( 0.99 - 0.16 ( c - 11 ) ) ( 0.5 + 0.09 ( c - 11 ) ) P(E=\bar{G}\bar{D}\mid C=c)=(0.99-0.16(c-11))(0.5+0.09(c-11))
  109. f C ( c ) = 0.2 \textstyle f_{C}(c)=0.2
  110. e e
  111. c c
  112. f C ( c E = e ) = P ( E = e C = c ) P ( E = e ) f C ( c ) = P ( E = e C = c ) 11 16 P ( E = e C = c ) f C ( c ) d c f C ( c ) f_{C}(c\mid E=e)=\frac{P(E=e\mid C=c)}{P(E=e)}f_{C}(c)=\frac{P(E=e\mid C=c)}{% \int_{11}^{16}{P(E=e\mid C=c)f_{C}(c)dc}}f_{C}(c)
  113. c = 15.2 c=15.2
  114. { G D , G D ¯ , G ¯ D , G ¯ D ¯ } \{GD,G\bar{D},\bar{G}D,\bar{G}\bar{D}\}

BCH_code.html

  1. q q
  2. m m
  3. d d
  4. G F ( q ) GF(q)
  5. d d
  6. α α
  7. i i
  8. G F ( q ) GF(q)
  9. g ( x ) g(x)
  10. G F ( q ) GF(q)
  11. g ( x ) g(x)
  12. q = 2 q=2
  13. m = 4 m=4
  14. n = 15 n=15
  15. d d
  16. α α
  17. G F ( 16 ) GF(16)
  18. G F ( 2 ) GF(2)
  19. m 1 ( x ) = x 4 + x + 1. m_{1}(x)=x^{4}+x+1.
  20. α α
  21. m 1 ( x ) = m 2 ( x ) = m 4 ( x ) = x 4 + x + 1 , m_{1}(x)=m_{2}(x)=m_{4}(x)=x^{4}+x+1,\,
  22. m 3 ( x ) = m 6 ( x ) = x 4 + x 3 + x 2 + x + 1 , m_{3}(x)=m_{6}(x)=x^{4}+x^{3}+x^{2}+x+1,\,
  23. m 5 ( x ) = x 2 + x + 1 , m_{5}(x)=x^{2}+x+1,\,
  24. m 7 ( x ) = x 4 + x 3 + 1. m_{7}(x)=x^{4}+x^{3}+1.\,
  25. d = 2 , 3 d=2,3
  26. g ( x ) = m 1 ( x ) = x 4 + x + 1. g(x)=m_{1}(x)=x^{4}+x+1.\,
  27. d = 4 , 5 d=4,5
  28. g ( x ) = lcm ( m 1 ( x ) , m 3 ( x ) ) = ( x 4 + x + 1 ) ( x 4 + x 3 + x 2 + x + 1 ) = x 8 + x 7 + x 6 + x 4 + 1. g(x)={\rm lcm}(m_{1}(x),m_{3}(x))=(x^{4}+x+1)(x^{4}+x^{3}+x^{2}+x+1)=x^{8}+x^{% 7}+x^{6}+x^{4}+1.\,
  29. d = 8 d=8
  30. g ( x ) = lcm ( m 1 ( x ) , m 3 ( x ) , m 5 ( x ) , m 7 ( x ) ) = ( x 4 + x + 1 ) ( x 4 + x 3 + x 2 + x + 1 ) ( x 2 + x + 1 ) ( x 4 + x 3 + 1 ) = x 14 + x 13 + x 12 + + x 2 + x + 1. \begin{aligned}\displaystyle g(x)&\displaystyle{}={\rm lcm}(m_{1}(x),m_{3}(x),% m_{5}(x),m_{7}(x))\\ &\displaystyle{}=(x^{4}+x+1)(x^{4}+x^{3}+x^{2}+x+1)(x^{2}+x+1)(x^{4}+x^{3}+1)% \\ &\displaystyle{}=x^{14}+x^{13}+x^{12}+\cdots+x^{2}+x+1.\end{aligned}
  31. α \alpha
  32. GF ( q m ) \mathrm{GF}(q^{m})
  33. q m - 1 q^{m}-1
  34. ord ( α ) , \mathrm{ord}(\alpha),
  35. α . \alpha.
  36. α c , , α c + d - 2 \alpha^{c},\ldots,\alpha^{c+d-2}
  37. α , , α d - 1 . \alpha,\ldots,\alpha^{d-1}.
  38. G F ( q ) , GF(q),
  39. q q
  40. m , n , d , c m,n,d,c
  41. 2 d n , 2\leq d\leq n,
  42. gcd ( n , q ) = 1 , {\rm gcd}(n,q)=1,
  43. m m
  44. q q
  45. n . n.
  46. α \alpha
  47. n n
  48. G F ( q m ) , GF(q^{m}),
  49. m i ( x ) m_{i}(x)
  50. G F ( q ) GF(q)
  51. α i \alpha^{i}
  52. i . i.
  53. g ( x ) = lcm ( m c ( x ) , , m c + d - 2 ( x ) ) . g(x)={\rm lcm}(m_{c}(x),\ldots,m_{c+d-2}(x)).
  54. n = q m - 1 n=q^{m}-1
  55. gcd ( n , q ) {\rm gcd}(n,q)
  56. q q
  57. n n
  58. m . m.
  59. c = 1 c=1
  60. n = q m - 1 n=q^{m}-1
  61. g ( x ) g(x)
  62. GF ( q ) . \mathrm{GF}(q).
  63. GF ( q p ) \mathrm{GF}(q^{p})
  64. g ( x ) g(x)
  65. GF ( q p ) . \mathrm{GF}(q^{p}).
  66. GF ( q m ) \mathrm{GF}(q^{m})
  67. g ( x ) g(x)
  68. ( d - 1 ) m . (d-1)m.
  69. q = 2 q=2
  70. c = 1 , c=1,
  71. d m / 2. dm/2.
  72. m i ( x ) m_{i}(x)
  73. m . m.
  74. d - 1 d-1
  75. ( d - 1 ) m . (d-1)m.
  76. q = 2 , q=2,
  77. m i ( x ) = m 2 i ( x ) m_{i}(x)=m_{2i}(x)
  78. i . i.
  79. g ( x ) g(x)
  80. d / 2 d/2
  81. m i ( x ) m_{i}(x)
  82. i , i,
  83. m . m.
  84. d . d.
  85. p ( x ) p(x)
  86. d d
  87. p ( x ) = b 1 x k 1 + + b d - 1 x k d - 1 , where k 1 < k 2 < < k d - 1 . p(x)=b_{1}x^{k_{1}}+\cdots+b_{d-1}x^{k_{d-1}},\,\text{ where }k_{1}<k_{2}<% \cdots<k_{d-1}.
  88. α c , , α c + d - 2 \alpha^{c},\ldots,\alpha^{c+d-2}
  89. g ( x ) , g(x),
  90. p ( x ) . p(x).
  91. b 1 , , b d - 1 b_{1},\ldots,b_{d-1}
  92. i = c , , c + d - 2 : i=c,\ldots,c+d-2:
  93. p ( α i ) = b 1 α i k 1 + b 2 α i k 2 + + b d - 1 α i k d - 1 = 0. p(\alpha^{i})=b_{1}\alpha^{ik_{1}}+b_{2}\alpha^{ik_{2}}+\cdots+b_{d-1}\alpha^{% ik_{d-1}}=0.
  94. [ α c k 1 α c k 2 α c k d - 1 α ( c + 1 ) k 1 α ( c + 1 ) k 2 α ( c + 1 ) k d - 1 α ( c + d - 2 ) k 1 α ( c + d - 2 ) k 2 α ( c + d - 2 ) k d - 1 ] [ b 1 b 2 b d - 1 ] = [ 0 0 0 ] . \begin{bmatrix}\alpha^{ck_{1}}&\alpha^{ck_{2}}&\cdots&\alpha^{ck_{d-1}}\\ \alpha^{(c+1)k_{1}}&\alpha^{(c+1)k_{2}}&\cdots&\alpha^{(c+1)k_{d-1}}\\ \vdots&\vdots&&\vdots\\ \alpha^{(c+d-2)k_{1}}&\alpha^{(c+d-2)k_{2}}&\cdots&\alpha^{(c+d-2)k_{d-1}}\\ \end{bmatrix}\begin{bmatrix}b_{1}\\ b_{2}\\ \vdots\\ b_{d-1}\end{bmatrix}=\begin{bmatrix}0\\ 0\\ \vdots\\ 0\end{bmatrix}.
  95. ( i = 1 d - 1 α c k i ) det ( 1 1 1 α k 1 α k 2 α k d - 1 α ( d - 2 ) k 1 α ( d - 2 ) k 2 α ( d - 2 ) k d - 1 ) = ( i = 1 d - 1 α c k i ) det ( V ) . \left(\prod_{i=1}^{d-1}\alpha^{ck_{i}}\right)\det\begin{pmatrix}1&1&\cdots&1\\ \alpha^{k_{1}}&\alpha^{k_{2}}&\cdots&\alpha^{k_{d-1}}\\ \vdots&\vdots&&\vdots\\ \alpha^{(d-2)k_{1}}&\alpha^{(d-2)k_{2}}&\cdots&\alpha^{(d-2)k_{d-1}}\\ \end{pmatrix}=\left(\prod_{i=1}^{d-1}\alpha^{ck_{i}}\right)\det(V).
  96. V V
  97. det ( V ) = 1 i < j d - 1 ( α k j - α k i ) , \det(V)=\prod_{1\leq i<j\leq d-1}(\alpha^{k_{j}}-\alpha^{k_{i}}),
  98. b 1 , , b d - 1 = 0 , b_{1},\ldots,b_{d-1}=0,
  99. p ( x ) = 0. p(x)=0.
  100. n n
  101. x n - 1. x^{n}-1.
  102. g ( x ) g(x)
  103. α c , , α c + d - 2 , \alpha^{c},\ldots,\alpha^{c+d-2},
  104. α c , , α c + d - 2 \alpha^{c},\ldots,\alpha^{c+d-2}
  105. x n - 1. x^{n}-1.
  106. α \alpha
  107. n n
  108. R R
  109. C C
  110. E . E.
  111. R R
  112. α c , , α c + d - 2 . \alpha^{c},\ldots,\alpha^{c+d-2}.
  113. s j = R ( α j ) = C ( α j ) + E ( α j ) s_{j}=R(\alpha^{j})=C(\alpha^{j})+E(\alpha^{j})
  114. j = c j=c
  115. c + d - 2. c+d-2.
  116. α j \alpha^{j}
  117. g ( x ) , g(x),
  118. C ( x ) C(x)
  119. C ( α j ) = 0. C(\alpha^{j})=0.
  120. s j = 0 s_{j}=0
  121. j . j.
  122. E ( x ) = e x i , E(x)=e\,x^{i},
  123. i i
  124. e e
  125. s c = e α c i s_{c}=e\,\alpha^{c\,i}
  126. s c + 1 = e α ( c + 1 ) i = α i s c s_{c+1}=e\,\alpha^{(c+1)\,i}=\alpha^{i}s_{c}
  127. e e
  128. i i
  129. E ( x ) = e 1 x i 1 + e 2 x i 2 + E(x)=e_{1}x^{i_{1}}+e_{2}x^{i_{2}}+\cdots\,
  130. e k e_{k}
  131. i k . i_{k}.
  132. Λ ( x ) = j = 1 t ( x α i j - 1 ) \Lambda(x)=\prod_{j=1}^{t}(x\alpha^{i_{j}}-1)
  133. t . t.
  134. λ 1 , λ 2 , , λ v \lambda_{1},\lambda_{2},\dots,\lambda_{v}
  135. Λ ( x ) = 1 + λ 1 x + λ 2 x 2 + + λ v x v . \Lambda(x)=1+\lambda_{1}x+\lambda_{2}x^{2}+\cdots+\lambda_{v}x^{v}.
  136. S v × v S_{v\times v}
  137. S v × v = [ s c s c + 1 s c + v - 1 s c + 1 s c + 2 s c + v s c + v - 1 s c + v s c + 2 v - 2 ] . S_{v\times v}=\begin{bmatrix}s_{c}&s_{c+1}&\dots&s_{c+v-1}\\ s_{c+1}&s_{c+2}&\dots&s_{c+v}\\ \vdots&\vdots&\ddots&\vdots\\ s_{c+v-1}&s_{c+v}&\dots&s_{c+2v-2}\end{bmatrix}.
  138. c v × 1 c_{v\times 1}
  139. C v × 1 = [ s c + v s c + v + 1 s c + 2 v - 1 ] . C_{v\times 1}=\begin{bmatrix}s_{c+v}\\ s_{c+v+1}\\ \vdots\\ s_{c+2v-1}\end{bmatrix}.
  140. Λ \Lambda
  141. Λ v × 1 = [ λ v λ v - 1 λ 1 ] . \Lambda_{v\times 1}=\begin{bmatrix}\lambda_{v}\\ \lambda_{v-1}\\ \vdots\\ \lambda_{1}\end{bmatrix}.
  142. S v × v Λ v × 1 = - C v × 1 . S_{v\times v}\Lambda_{v\times 1}=-C_{v\times 1\,}.
  143. S v × v S_{v\times v}
  144. Λ \Lambda
  145. det ( S v × v ) = 0 , \det(S_{v\times v})=0,
  146. v = 0 v=0
  147. v v - 1 v\leftarrow v-1
  148. S v × v S_{v\times v}
  149. Λ \Lambda
  150. Λ ( x ) \Lambda(x)
  151. Λ ( x ) = ( α i 1 x - 1 ) ( α i 2 x - 1 ) ( α i v x - 1 ) \Lambda(x)=(\alpha^{i_{1}}x-1)(\alpha^{i_{2}}x-1)\cdots(\alpha^{i_{v}}x-1)
  152. α \alpha
  153. e j e_{j}
  154. s c \displaystyle s_{c}
  155. S ( x ) = s c + s c + 1 x + s c + 2 x 2 + + s c + d - 2 x d - 2 . S(x)=s_{c}+s_{c+1}x+s_{c+2}x^{2}+\cdots+s_{c+d-2}x^{d-2}.
  156. v d - 1 , v\leq d-1,
  157. λ 0 0 , \lambda_{0}\neq 0,
  158. Λ ( x ) = i = 0 v λ i x i = λ 0 k = 0 v ( α - i k x - 1 ) . \Lambda(x)=\sum_{i=0}^{v}\lambda_{i}x^{i}=\lambda_{0}\cdot\prod_{k=0}^{v}(% \alpha^{-i_{k}}x-1).
  159. Ω ( x ) = S ( x ) Λ ( x ) ( mod x d - 1 ) \Omega(x)=S(x)\,\Lambda(x)\;\;(\mathop{{\rm mod}}x^{d-1})
  160. Λ ( x ) = Σ i = 1 v i λ i x i - 1 , \Lambda^{\prime}(x)=\Sigma_{i=1}^{v}i\cdot\lambda_{i}x^{i-1},
  161. i x i\cdot x
  162. k = 1 i x \textstyle\sum_{k=1}^{i}x
  163. i k i_{k}
  164. e k = - α i k Ω ( α - i k ) α c i k Λ ( α - i k ) . e_{k}=-{\alpha^{i_{k}}\Omega(\alpha^{-i_{k}})\over\alpha^{c\cdot i_{k}}\Lambda% ^{\prime}(\alpha^{-i_{k}})}.
  165. e k = - Ω ( α - i k ) Λ ( α - i k ) . e_{k}=-{\Omega(\alpha^{-i_{k}})\over\Lambda^{\prime}(\alpha^{-i_{k}})}.
  166. S ( x ) Λ ( x ) . S(x)\Lambda(x).
  167. λ k = 0 \lambda_{k}=0
  168. k > v , k>v,
  169. s k = 0 s_{k}=0
  170. k > c + d - 2. k>c+d-2.
  171. S ( x ) Λ ( x ) = j = 0 i = 0 j s j - i + 1 λ i x j . S(x)\Lambda(x)=\sum_{j=0}^{\infty}\sum_{i=0}^{j}s_{j-i+1}\lambda_{i}x^{j}.
  172. S ( x ) = i = 0 d - 2 j = 1 v e j α ( c + i ) i j x i = j = 1 v e j α c i j i = 0 d - 2 ( α i j ) i x i = j = 1 v e j α c i j ( x α i j ) d - 1 - 1 x α i j - 1 . S(x)=\sum_{i=0}^{d-2}\sum_{j=1}^{v}e_{j}\alpha^{(c+i)\cdot i_{j}}x^{i}=\sum_{j% =1}^{v}e_{j}\alpha^{c\,i_{j}}\sum_{i=0}^{d-2}(\alpha^{i_{j}})^{i}x^{i}=\sum_{j% =1}^{v}e_{j}\alpha^{c\,i_{j}}{(x\alpha^{i_{j}})^{d-1}-1\over x\alpha^{i_{j}}-1}.
  173. S ( x ) Λ ( x ) = S ( x ) λ 0 = 1 v ( α i x - 1 ) = λ 0 j = 1 v e j α c i j ( x α i j ) d - 1 - 1 x α i j - 1 = 1 v ( α i x - 1 ) . S(x)\Lambda(x)=S(x)\lambda_{0}\prod_{\ell=1}^{v}(\alpha^{i_{\ell}}x-1)=\lambda% _{0}\sum_{j=1}^{v}e_{j}\alpha^{c\,i_{j}}{(x\alpha^{i_{j}})^{d-1}-1\over x% \alpha^{i_{j}}-1}\prod_{\ell=1}^{v}(\alpha^{i_{\ell}}x-1).
  174. S ( x ) Λ ( x ) = λ 0 j = 1 v e j α c i j ( ( x α i j ) d - 1 - 1 ) { 1 , , v } { j } ( α i x - 1 ) . S(x)\Lambda(x)=\lambda_{0}\sum_{j=1}^{v}e_{j}\alpha^{c\,i_{j}}((x\alpha^{i_{j}% })^{d-1}-1)\prod_{\ell\in\{1,\dots,v\}\setminus\{j\}}(\alpha^{i_{\ell}}x-1).
  175. e j , e_{j},
  176. ( x α i j ) d - 1 (x\alpha^{i_{j}})^{d-1}
  177. Ω ( x ) = S ( x ) Λ ( x ) ( mod x d - 1 ) . \Omega(x)=S(x)\,\Lambda(x)\;\;(\mathop{{\rm mod}}x^{d-1}).
  178. v d - 1 v\leq d-1
  179. Ω ( x ) = - λ 0 j = 1 v e j α c i j { 1 , , v } { j } ( α i x - 1 ) . \Omega(x)=-\lambda_{0}\sum_{j=1}^{v}e_{j}\alpha^{c\,i_{j}}\prod_{\ell\in\{1,% \dots,v\}\setminus\{j\}}(\alpha^{i_{\ell}}x-1).
  180. Ω ( α - i k ) . \Omega(\alpha^{-i_{k}}).
  181. Λ \Lambda
  182. Ω ( α - i k ) = - λ 0 e k α c i k { 1 , , v } { k } ( α i α - i k - 1 ) . \Omega(\alpha^{-i_{k}})=-\lambda_{0}e_{k}\alpha^{c\cdot i_{k}}\prod_{\ell\in\{% 1,\dots,v\}\setminus\{k\}}(\alpha^{i_{\ell}}\alpha^{-i_{k}}-1).
  183. e k e_{k}
  184. α - i j \alpha^{-i_{j}}
  185. Λ , \Lambda,
  186. Λ ( x ) = λ 0 j = 1 v α i j { 1 , , v } { j } ( α i x - 1 ) , \Lambda^{\prime}(x)=\lambda_{0}\sum_{j=1}^{v}\alpha^{i_{j}}\prod_{\ell\in\{1,% \dots,v\}\setminus\{j\}}(\alpha^{i_{\ell}}x-1),
  187. Λ ( α - i k ) = λ 0 α i k { 1 , , v } { k } ( α i α - i k - 1 ) . \Lambda^{\prime}(\alpha^{-i_{k}})=\lambda_{0}\alpha^{i_{k}}\prod_{\ell\in\{1,% \dots,v\}\setminus\{k\}}(\alpha^{i_{\ell}}\alpha^{-i_{k}}-1).
  188. e k = - α i k Ω ( α - i k ) α c i k Λ ( α - i k ) . e_{k}=-{\alpha^{i_{k}}\Omega(\alpha^{-i_{k}})\over\alpha^{c\cdot i_{k}}\Lambda% ^{\prime}(\alpha^{-i_{k}})}.
  189. Λ \Lambda
  190. Λ ( x ) = i = 1 v λ i x i \Lambda(x)=\sum_{i=1}^{v}\lambda_{i}x^{i}
  191. Λ ( x ) = Σ i = 1 v i λ i x i - 1 , \Lambda^{\prime}(x)=\Sigma_{i=1}^{v}i\cdot\lambda_{i}x^{i-1},
  192. i x i\cdot x
  193. k = 1 i x \textstyle\sum_{k=1}^{i}x
  194. k 1 , , k k k_{1},...,k_{k}
  195. Γ ( x ) = i = 1 k ( x α k i - 1 ) . \Gamma(x)=\prod_{i=1}^{k}(x\alpha^{k_{i}}-1).
  196. S ( x ) = i = 0 d - 2 s c + i x i . S(x)=\sum_{i=0}^{d-2}s_{c+i}x^{i}.
  197. S ( x ) Γ ( x ) S(x)\Gamma(x)
  198. x d - 1 . x^{d-1}.
  199. r ( x ) r(x)
  200. ( d + k - 3 ) / 2 \lfloor(d+k-3)/2\rfloor
  201. a ( x ) , b ( x ) a(x),b(x)
  202. r ( x ) = a ( x ) S ( x ) Γ ( x ) + b ( x ) x d - 1 . r(x)=a(x)S(x)\Gamma(x)+b(x)x^{d-1}.
  203. r ( x ) r(x)
  204. a ( x ) a(x)
  205. Γ \Gamma
  206. Λ . \Lambda.
  207. Ξ ( x ) = a ( x ) Γ ( x ) \Xi(x)=a(x)\Gamma(x)
  208. Ξ \Xi
  209. Λ ( x ) \Lambda(x)
  210. Ω ( x ) = S ( x ) Ξ ( x ) mod x d - 1 = r ( x ) \Omega(x)=S(x)\Xi(x)\bmod x^{d-1}=r(x)
  211. s i s_{i}
  212. s i = j = 0 n - 1 e j α i j . s_{i}=\sum_{j=0}^{n-1}e_{j}\alpha^{ij}.
  213. S ( x ) = i = 0 d - 2 s c + i x i S(x)=\sum_{i=0}^{d-2}s_{c+i}x^{i}
  214. 0
  215. d - 2. d-2.
  216. S ( x ) { 0 , , d - 2 } = E ( x ) = i = 0 d - 2 j = 0 n - 1 e j α i j α c j x i . S(x){\textstyle{\{0,\ldots,\,d-2\}\atop=}}E(x)=\sum_{i=0}^{d-2}\sum_{j=0}^{n-1% }e_{j}\alpha^{ij}\alpha^{cj}x^{i}.
  217. k 1 , k_{1},
  218. { s c , , s c + d - 2 } \{s_{c},\ldots,s_{c+d-2}\}
  219. { t c , , t c + d - 3 } \{t_{c},\ldots,t_{c+d-3}\}
  220. t i = α k 1 s i - s i + 1 . t_{i}=\alpha^{k_{1}}s_{i}-s_{i+1}.
  221. { s c , , s c + d - 2 } \{s_{c},\ldots,s_{c+d-2}\}
  222. t i = α k 1 s i - s i + 1 = α k 1 j = 0 n - 1 e j α i j - j = 0 n - 1 e j α j α i j = j = 0 n - 1 e j ( α k 1 - α j ) α i j . t_{i}=\alpha^{k_{1}}s_{i}-s_{i+1}=\alpha^{k_{1}}\sum_{j=0}^{n-1}e_{j}\alpha^{% ij}-\sum_{j=0}^{n-1}e_{j}\alpha^{j}\alpha^{ij}=\sum_{j=0}^{n-1}e_{j}(\alpha^{k% _{1}}-\alpha^{j})\alpha^{ij}.
  223. f j = e j ( α k 1 - α j ) f_{j}=e_{j}(\alpha^{k_{1}}-\alpha^{j})
  224. e j . e_{j}.
  225. k 1 , k_{1},
  226. f k 1 = 0 , f_{k_{1}}=0,
  227. f j f_{j}
  228. e j e_{j}
  229. k . k.
  230. { s c , , s c + d - 2 } \{s_{c},\ldots,s_{c+d-2}\}
  231. { t c , , t c + d - 3 } \{t_{c},\ldots,t_{c+d-3}\}
  232. T ( x ) = i = 0 d - 3 t c + i x i = α k 1 i = 0 d - 3 s c + i x i - i = 1 d - 2 s c + i x i - 1 . T(x)=\sum_{i=0}^{d-3}t_{c+i}x^{i}=\alpha^{k_{1}}\sum_{i=0}^{d-3}s_{c+i}x^{i}-% \sum_{i=1}^{d-2}s_{c+i}x^{i-1}.
  233. x T ( x ) { 1 , , d - 2 } = ( x α k 1 - 1 ) S ( x ) . xT(x){\textstyle{\{1,\ldots,\,d-2\}\atop=}}(x\alpha^{k_{1}}-1)S(x).
  234. S ( x ) S(x)
  235. S ( x ) Γ ( x ) S(x)\Gamma(x)
  236. k , , d - 2. k,\ldots,d-2.
  237. v v
  238. Λ ( x ) \Lambda(x)
  239. S ( x ) Γ ( x ) Λ ( x ) { k + v , , d - 2 } = 0. S(x)\Gamma(x)\Lambda(x){\textstyle{\{k+v,\ldots,\,d-2\}\atop=}}0.
  240. ( d - 1 - k ) / 2 (d-1-k)/2
  241. Λ ( x ) \Lambda(x)
  242. k + ( d - 1 - k ) / 2 . k+\lfloor(d-1-k)/2\rfloor.
  243. Λ ( x ) \Lambda(x)
  244. Λ ( x ) \Lambda(x)
  245. ( d - 1 - k ) / 2 , (d-1-k)/2,
  246. Λ ( x ) \Lambda(x)
  247. Λ ( x ) \Lambda(x)
  248. d = 7 d=7
  249. g ( x ) = x 10 + x 8 + x 5 + x 4 + x 2 + x + 1 g(x)=x^{10}+x^{8}+x^{5}+x^{4}+x^{2}+x+1
  250. M ( x ) = x 4 + x 3 + x + 1. M(x)=x^{4}+x^{3}+x+1.
  251. x 10 M ( x ) x^{10}M(x)
  252. g ( x ) g(x)
  253. x 9 + x 4 + x 2 x^{9}+x^{4}+x^{2}
  254. R ( x ) = C ( x ) + x 13 + x 5 = x 14 + x 11 + x 10 + x 9 + x 5 + x 4 + x 2 R(x)=C(x)+x^{13}+x^{5}=x^{14}+x^{11}+x^{10}+x^{9}+x^{5}+x^{4}+x^{2}
  255. α = 0010 , \alpha=0010,
  256. s 1 = R ( α 1 ) = 1011 , s_{1}=R(\alpha^{1})=1011,
  257. s 2 = 1001 , s_{2}=1001,
  258. s 3 = 1011 , s_{3}=1011,
  259. s 4 = 1101 , s_{4}=1101,
  260. s 5 = 0001 , s_{5}=0001,
  261. s 6 = 1001. s_{6}=1001.
  262. [ S 3 × 3 | C 3 × 1 ] = [ s 1 s 2 s 3 s 4 s 2 s 3 s 4 s 5 s 3 s 4 s 5 s 6 ] = [ 1011 1001 1011 1101 1001 1011 1101 0001 1011 1101 0001 1001 ] [ 0001 0000 1000 0111 0000 0001 1011 0001 0000 0000 0000 0000 ] \left[S_{3\times 3}|C_{3\times 1}\right]=\begin{bmatrix}s_{1}&s_{2}&s_{3}&s_{4% }\\ s_{2}&s_{3}&s_{4}&s_{5}\\ s_{3}&s_{4}&s_{5}&s_{6}\end{bmatrix}=\begin{bmatrix}1011&1001&1011&1101\\ 1001&1011&1101&0001\\ 1011&1101&0001&1001\end{bmatrix}\Rightarrow\begin{bmatrix}0001&0000&1000&0111% \\ 0000&0001&1011&0001\\ 0000&0000&0000&0000\end{bmatrix}
  263. λ 2 = 1000 , \lambda_{2}=1000,
  264. λ 1 = 1011. \lambda_{1}=1011.
  265. Λ ( x ) = 1000 x 2 + 1011 x + 0001 , \Lambda(x)=1000x^{2}+1011x+0001,
  266. 0100 = α - 13 0100=\alpha^{-13}
  267. 0111 = α - 5 . 0111=\alpha^{-5}.
  268. α \alpha
  269. Γ ( x ) = ( α 8 x - 1 ) ( α 11 x - 1 ) . \Gamma(x)=(\alpha^{8}x-1)(\alpha^{11}x-1).
  270. s 1 = α - 7 , s_{1}=\alpha^{-7},
  271. s 2 = α 1 , s_{2}=\alpha^{1},
  272. s 3 = α 4 , s_{3}=\alpha^{4},
  273. s 4 = α 2 , s_{4}=\alpha^{2},
  274. s 5 = α 5 , s_{5}=\alpha^{5},
  275. s 6 = α - 7 . s_{6}=\alpha^{-7}.
  276. α \alpha
  277. S ( x ) = α - 7 + α 1 x + α 4 x 2 + α 2 x 3 + α 5 x 4 + α - 7 x 5 , S(x)=\alpha^{-7}+\alpha^{1}x+\alpha^{4}x^{2}+\alpha^{2}x^{3}+\alpha^{5}x^{4}+% \alpha^{-7}x^{5},
  278. S ( x ) Γ ( x ) = α - 7 + α 4 x + α - 1 x 2 + α 6 x 3 + α - 1 x 4 + α 5 x 5 + α 7 x 6 + α - 3 x 7 . S(x)\Gamma(x)=\alpha^{-7}+\alpha^{4}x+\alpha^{-1}x^{2}+\alpha^{6}x^{3}+\alpha^% {-1}x^{4}+\alpha^{5}x^{5}+\alpha^{7}x^{6}+\alpha^{-3}x^{7}.
  279. ( S ( x ) Γ ( x ) x 6 ) = ( α - 7 + α 4 x + α - 1 x 2 + α 6 x 3 + α - 1 x 4 + α 5 x 5 + α 7 x 6 + α - 3 x 7 x 6 ) \begin{pmatrix}S(x)\Gamma(x)\\ x^{6}\end{pmatrix}=\begin{pmatrix}\alpha^{-7}+\alpha^{4}x+\alpha^{-1}x^{2}+% \alpha^{6}x^{3}+\alpha^{-1}x^{4}+\alpha^{5}x^{5}+\alpha^{7}x^{6}+\alpha^{-3}x^% {7}\\ x^{6}\end{pmatrix}
  280. = ( α 7 + α - 3 x 1 1 0 ) ( x 6 α - 7 + α 4 x + α - 1 x 2 + α 6 x 3 + α - 1 x 4 + α 5 x 5 + ( α 7 + α 7 ) x 6 + ( α - 3 + α - 3 ) x 7 ) =\begin{pmatrix}\alpha^{7}+\alpha^{-3}x&1\\ 1&0\end{pmatrix}\begin{pmatrix}x^{6}\\ \alpha^{-7}+\alpha^{4}x+\alpha^{-1}x^{2}+\alpha^{6}x^{3}+\alpha^{-1}x^{4}+% \alpha^{5}x^{5}+(\alpha^{7}+\alpha^{7})x^{6}+(\alpha^{-3}+\alpha^{-3})x^{7}% \end{pmatrix}
  281. = ( α 7 + α - 3 x 1 1 0 ) ( α 4 + α - 5 x 1 1 0 ) ( α - 7 + α 4 x + α - 1 x 2 + α 6 x 3 + α - 1 x 4 + α 5 x 5 α - 3 + ( α - 7 + α 3 ) x + ( α 3 + α - 1 ) x 2 + ( α - 5 + α - 6 ) x 3 + ( α 3 + α 1 ) x 4 + ( α - 6 + α - 6 ) x 5 + ( α 0 + 1 ) x 6 ) =\begin{pmatrix}\alpha^{7}+\alpha^{-3}x&1\\ 1&0\end{pmatrix}\begin{pmatrix}\alpha^{4}+\alpha^{-5}x&1\\ 1&0\end{pmatrix}\begin{pmatrix}\alpha^{-7}+\alpha^{4}x+\alpha^{-1}x^{2}+\alpha% ^{6}x^{3}+\alpha^{-1}x^{4}+\alpha^{5}x^{5}\\ \alpha^{-3}+(\alpha^{-7}+\alpha^{3})x+(\alpha^{3}+\alpha^{-1})x^{2}+\\ (\alpha^{-5}+\alpha^{-6})x^{3}+(\alpha^{3}+\alpha^{1})x^{4}+(\alpha^{-6}+% \alpha^{-6})x^{5}+(\alpha^{0}+1)x^{6}\end{pmatrix}
  282. = ( ( 1 + α - 4 ) + ( α 1 + α 2 ) x + α 7 x 2 α 7 + α - 3 x α 4 + α - 5 x 1 ) ( α - 7 + α 4 x + α - 1 x 2 + α 6 x 3 + α - 1 x 4 + α 5 x 5 α - 3 + α - 2 x + α 0 x 2 + α - 2 x 3 + α - 6 x 4 ) =\begin{pmatrix}(1+\alpha^{-4})+(\alpha^{1}+\alpha^{2})x+\alpha^{7}x^{2}&% \alpha^{7}+\alpha^{-3}x\\ \alpha^{4}+\alpha^{-5}x&1\end{pmatrix}\begin{pmatrix}\alpha^{-7}+\alpha^{4}x+% \alpha^{-1}x^{2}+\alpha^{6}x^{3}+\alpha^{-1}x^{4}+\alpha^{5}x^{5}\\ \alpha^{-3}+\alpha^{-2}x+\alpha^{0}x^{2}+\alpha^{-2}x^{3}+\alpha^{-6}x^{4}\end% {pmatrix}
  283. = ( α - 3 + α 5 x + α 7 x 2 α 7 + α - 3 x α 4 + α - 5 x 1 ) ( α - 5 + α - 4 x 1 1 0 ) ( α - 3 + α - 2 x + α 0 x 2 + α - 2 x 3 + α - 6 x 4 ( α 7 + α - 7 ) + ( α - 7 + α - 7 + α 4 ) x + ( α - 5 + α - 6 + α - 1 ) x 2 + ( α - 7 + α - 4 + α 6 ) x 3 + ( α 4 + α - 6 + α - 1 ) x 4 + ( α 5 + α 5 ) x 5 ) =\begin{pmatrix}\alpha^{-3}+\alpha^{5}x+\alpha^{7}x^{2}&\alpha^{7}+\alpha^{-3}% x\\ \alpha^{4}+\alpha^{-5}x&1\end{pmatrix}\begin{pmatrix}\alpha^{-5}+\alpha^{-4}x&% 1\\ 1&0\end{pmatrix}\begin{pmatrix}\alpha^{-3}+\alpha^{-2}x+\alpha^{0}x^{2}+\alpha% ^{-2}x^{3}+\alpha^{-6}x^{4}\\ (\alpha^{7}+\alpha^{-7})+(\alpha^{-7}+\alpha^{-7}+\alpha^{4})x+\\ (\alpha^{-5}+\alpha^{-6}+\alpha^{-1})x^{2}+\\ (\alpha^{-7}+\alpha^{-4}+\alpha^{6})x^{3}+\\ (\alpha^{4}+\alpha^{-6}+\alpha^{-1})x^{4}+(\alpha^{5}+\alpha^{5})x^{5}\end{pmatrix}
  284. = ( α 7 x + α 5 x 2 + α 3 x 3 α - 3 + α 5 x + α 7 x 2 α 3 + α - 5 x + α 6 x 2 α 4 + α - 5 x ) ( α - 3 + α - 2 x + α 0 x 2 + α - 2 x 3 + α - 6 x 4 α - 4 + α 4 x + α 2 x 2 + α - 5 x 3 ) . =\begin{pmatrix}\alpha^{7}x+\alpha^{5}x^{2}+\alpha^{3}x^{3}&\alpha^{-3}+\alpha% ^{5}x+\alpha^{7}x^{2}\\ \alpha^{3}+\alpha^{-5}x+\alpha^{6}x^{2}&\alpha^{4}+\alpha^{-5}x\end{pmatrix}% \begin{pmatrix}\alpha^{-3}+\alpha^{-2}x+\alpha^{0}x^{2}+\alpha^{-2}x^{3}+% \alpha^{-6}x^{4}\\ \alpha^{-4}+\alpha^{4}x+\alpha^{2}x^{2}+\alpha^{-5}x^{3}\end{pmatrix}.
  285. ( - ( α 4 + α - 5 x ) α - 3 + α 5 x + α 7 x 2 α 3 + α - 5 x + α 6 x 2 - ( α 7 x + α 5 x 2 + α 3 x 3 ) ) ( α 7 x + α 5 x 2 + α 3 x 3 α - 3 + α 5 x + α 7 x 2 α 3 + α - 5 x + α 6 x 2 α 4 + α - 5 x ) = ( 1 0 0 1 ) , \begin{pmatrix}-(\alpha^{4}+\alpha^{-5}x)&\alpha^{-3}+\alpha^{5}x+\alpha^{7}x^% {2}\\ \alpha^{3}+\alpha^{-5}x+\alpha^{6}x^{2}&-(\alpha^{7}x+\alpha^{5}x^{2}+\alpha^{% 3}x^{3})\end{pmatrix}\begin{pmatrix}\alpha^{7}x+\alpha^{5}x^{2}+\alpha^{3}x^{3% }&\alpha^{-3}+\alpha^{5}x+\alpha^{7}x^{2}\\ \alpha^{3}+\alpha^{-5}x+\alpha^{6}x^{2}&\alpha^{4}+\alpha^{-5}x\end{pmatrix}=% \begin{pmatrix}1&0\\ 0&1\end{pmatrix},
  286. ( - ( α 4 + α - 5 x ) α - 3 + α 5 x + α 7 x 2 α 3 + α - 5 x + α 6 x 2 - ( α 7 x + α 5 x 2 + α 3 x 3 ) ) ( S ( x ) Γ ( x ) x 6 ) = ( α - 3 + α - 2 x + α 0 x 2 + α - 2 x 3 + α - 6 x 4 α - 4 + α 4 x + α 2 x 2 + α - 5 x 3 ) . \begin{pmatrix}-(\alpha^{4}+\alpha^{-5}x)&\alpha^{-3}+\alpha^{5}x+\alpha^{7}x^% {2}\\ \alpha^{3}+\alpha^{-5}x+\alpha^{6}x^{2}&-(\alpha^{7}x+\alpha^{5}x^{2}+\alpha^{% 3}x^{3})\end{pmatrix}\begin{pmatrix}S(x)\Gamma(x)\\ x^{6}\end{pmatrix}=\begin{pmatrix}\alpha^{-3}+\alpha^{-2}x+\alpha^{0}x^{2}+\\ \alpha^{-2}x^{3}+\alpha^{-6}x^{4}\\ \alpha^{-4}+\alpha^{4}x+\alpha^{2}x^{2}+\\ \alpha^{-5}x^{3}\end{pmatrix}.
  287. S ( x ) Γ ( x ) ( α 3 + α - 5 x + α 6 x 2 ) - ( α 7 x + α 5 x 2 + α 3 x 3 ) x 6 = α - 4 + α 4 x + α 2 x 2 + α - 5 x 3 . S(x)\Gamma(x)(\alpha^{3}+\alpha^{-5}x+\alpha^{6}x^{2})-(\alpha^{7}x+\alpha^{5}% x^{2}+\alpha^{3}x^{3})x^{6}=\alpha^{-4}+\alpha^{4}x+\alpha^{2}x^{2}+\alpha^{-5% }x^{3}.
  288. Λ ( x ) = α 3 + α - 5 x + α 6 x 2 . \Lambda(x)=\alpha^{3}+\alpha^{-5}x+\alpha^{6}x^{2}.
  289. λ 0 1. \lambda_{0}\neq 1.
  290. Λ . \Lambda.
  291. α 2 , \alpha^{2},
  292. α 10 \alpha^{10}
  293. α 2 \alpha^{2}
  294. Λ \Lambda
  295. ( x - α 2 ) (x-\alpha^{2})
  296. Ξ ( x ) = Γ ( x ) Λ ( x ) = α 3 + α 4 x 2 + α 2 x 3 + α - 5 x 4 , \Xi(x)=\Gamma(x)\Lambda(x)=\alpha^{3}+\alpha^{4}x^{2}+\alpha^{2}x^{3}+\alpha^{% -5}x^{4},
  297. Ω ( x ) = S ( x ) Ξ ( x ) mod x 6 = α - 4 + α 4 x + α 2 x 2 + α - 5 x 3 . \Omega(x)=S(x)\Xi(x)\,\bmod\,x^{6}=\alpha^{-4}+\alpha^{4}x+\alpha^{2}x^{2}+% \alpha^{-5}x^{3}.
  298. e j = - Ω ( α - i j ) / Ξ ( α - i j ) , e_{j}=-\Omega(\alpha^{-i_{j}})/\Xi^{\prime}(\alpha^{-i_{j}}),
  299. α - i j \alpha^{-i_{j}}
  300. Ξ ( x ) . \Xi(x).
  301. Ξ ( x ) = α 2 x 2 . \Xi^{\prime}(x)=\alpha^{2}x^{2}.
  302. e 1 = - Ω ( α 4 ) / Ξ ( α 4 ) = ( α - 4 + α - 7 + α - 5 + α 7 ) / α - 5 = α - 5 / α - 5 = 1 , e_{1}=-\Omega(\alpha^{4})/\Xi^{\prime}(\alpha^{4})=(\alpha^{-4}+\alpha^{-7}+% \alpha^{-5}+\alpha^{7})/\alpha^{-5}=\alpha^{-5}/\alpha^{-5}=1,
  303. e 2 = - Ω ( α 7 ) / Ξ ( α 7 ) = ( α - 4 + α - 4 + α 1 + α 1 ) / α 1 = 0 , e_{2}=-\Omega(\alpha^{7})/\Xi^{\prime}(\alpha^{7})=(\alpha^{-4}+\alpha^{-4}+% \alpha^{1}+\alpha^{1})/\alpha^{1}=0,
  304. e 3 = - Ω ( α 10 ) / Ξ ( α 10 ) = ( α - 4 + α - 1 + α 7 + α - 5 ) / α 7 = α 7 / α 7 = 1 , e_{3}=-\Omega(\alpha^{10})/\Xi^{\prime}(\alpha^{10})=(\alpha^{-4}+\alpha^{-1}+% \alpha^{7}+\alpha^{-5})/\alpha^{7}=\alpha^{7}/\alpha^{7}=1,
  305. e 4 = - Ω ( α 2 ) / Ξ ( α 2 ) = ( α - 4 + α 6 + α 6 + α 1 ) / α 6 = α 6 / α 6 = 1. e_{4}=-\Omega(\alpha^{2})/\Xi^{\prime}(\alpha^{2})=(\alpha^{-4}+\alpha^{6}+% \alpha^{6}+\alpha^{1})/\alpha^{6}=\alpha^{6}/\alpha^{6}=1.
  306. e 3 = e 4 = 1 , e_{3}=e_{4}=1,
  307. Γ ( x ) = ( α 8 x - 1 ) ( α 11 x - 1 ) . \Gamma(x)=(\alpha^{8}x-1)(\alpha^{11}x-1).
  308. s 1 = α 4 , s_{1}=\alpha^{4},
  309. s 2 = α - 7 , s_{2}=\alpha^{-7},
  310. s 3 = α 1 , s_{3}=\alpha^{1},
  311. s 4 = α 1 , s_{4}=\alpha^{1},
  312. s 5 = α 0 , s_{5}=\alpha^{0},
  313. s 6 = α 2 . s_{6}=\alpha^{2}.
  314. S ( x ) = α 4 + α - 7 x + α 1 x 2 + α 1 x 3 + α 0 x 4 + α 2 x 5 , S(x)=\alpha^{4}+\alpha^{-7}x+\alpha^{1}x^{2}+\alpha^{1}x^{3}+\alpha^{0}x^{4}+% \alpha^{2}x^{5},
  315. S ( x ) Γ ( x ) = α 4 + α 7 x + α 5 x 2 + α 3 x 3 + α 1 x 4 + α - 1 x 5 + α - 1 x 6 + α 6 x 7 . S(x)\Gamma(x)=\alpha^{4}+\alpha^{7}x+\alpha^{5}x^{2}+\alpha^{3}x^{3}+\alpha^{1% }x^{4}+\alpha^{-1}x^{5}+\alpha^{-1}x^{6}+\alpha^{6}x^{7}.
  316. ( S ( x ) Γ ( x ) x 6 ) = ( α 4 + α 7 x + α 5 x 2 + α 3 x 3 + α 1 x 4 + α - 1 x 5 + α - 1 x 6 + α 6 x 7 x 6 ) \begin{pmatrix}S(x)\Gamma(x)\\ x^{6}\end{pmatrix}=\begin{pmatrix}\alpha^{4}+\alpha^{7}x+\alpha^{5}x^{2}+% \alpha^{3}x^{3}+\alpha^{1}x^{4}+\alpha^{-1}x^{5}+\alpha^{-1}x^{6}+\alpha^{6}x^% {7}\\ x^{6}\end{pmatrix}
  317. = ( α - 1 + α 6 x 1 1 0 ) ( x 6 α 4 + α 7 x + α 5 x 2 + α 3 x 3 + α 1 x 4 + α - 1 x 5 + ( α - 1 + α - 1 ) x 6 + ( α 6 + α 6 ) x 7 ) =\begin{pmatrix}\alpha^{-1}+\alpha^{6}x&1\\ 1&0\end{pmatrix}\begin{pmatrix}x^{6}\\ \alpha^{4}+\alpha^{7}x+\alpha^{5}x^{2}+\alpha^{3}x^{3}+\alpha^{1}x^{4}+\alpha^% {-1}x^{5}+(\alpha^{-1}+\alpha^{-1})x^{6}+(\alpha^{6}+\alpha^{6})x^{7}\end{pmatrix}
  318. = ( α - 1 + α 6 x 1 1 0 ) ( α 3 + α 1 x 1 1 0 ) ( α 4 + α 7 x + α 5 x 2 + α 3 x 3 + α 1 x 4 + α - 1 x 5 α 7 + ( α - 5 + α 5 ) x + ( α - 7 + α - 7 ) x 2 + ( α 6 + α 6 ) x 3 + ( α 4 + α 4 ) x 4 + ( α 2 + α 2 ) x 5 + ( α 0 + 1 ) x 6 ) =\begin{pmatrix}\alpha^{-1}+\alpha^{6}x&1\\ 1&0\end{pmatrix}\begin{pmatrix}\alpha^{3}+\alpha^{1}x&1\\ 1&0\end{pmatrix}\begin{pmatrix}\alpha^{4}+\alpha^{7}x+\alpha^{5}x^{2}+\alpha^{% 3}x^{3}+\alpha^{1}x^{4}+\alpha^{-1}x^{5}\\ \alpha^{7}+(\alpha^{-5}+\alpha^{5})x+(\alpha^{-7}+\alpha^{-7})x^{2}+(\alpha^{6% }+\alpha^{6})x^{3}+\\ (\alpha^{4}+\alpha^{4})x^{4}+(\alpha^{2}+\alpha^{2})x^{5}+(\alpha^{0}+1)x^{6}% \end{pmatrix}
  319. = ( ( 1 + α 2 ) + ( α 0 + α - 6 ) x + α 7 x 2 α - 1 + α 6 x α 3 + α 1 x 1 ) ( α 4 + α 7 x + α 5 x 2 + α 3 x 3 + α 1 x 4 + α - 1 x 5 α 7 + α 0 x ) . =\begin{pmatrix}(1+\alpha^{2})+(\alpha^{0}+\alpha^{-6})x+\alpha^{7}x^{2}&% \alpha^{-1}+\alpha^{6}x\\ \alpha^{3}+\alpha^{1}x&1\end{pmatrix}\begin{pmatrix}\alpha^{4}+\alpha^{7}x+% \alpha^{5}x^{2}+\alpha^{3}x^{3}+\alpha^{1}x^{4}+\alpha^{-1}x^{5}\\ \alpha^{7}+\alpha^{0}x\end{pmatrix}.
  320. ( - ( 1 ) α - 1 + α 6 x α 3 + α 1 x - ( α - 7 + α 7 x + α 7 x 2 ) ) ( α - 7 + α 7 x + α 7 x 2 α - 1 + α 6 x α 3 + α 1 x 1 ) = ( 1 0 0 1 ) , \begin{pmatrix}-(1)&\alpha^{-1}+\alpha^{6}x\\ \alpha^{3}+\alpha^{1}x&-(\alpha^{-7}+\alpha^{7}x+\alpha^{7}x^{2})\end{pmatrix}% \begin{pmatrix}\alpha^{-7}+\alpha^{7}x+\alpha^{7}x^{2}&\alpha^{-1}+\alpha^{6}x% \\ \alpha^{3}+\alpha^{1}x&1\end{pmatrix}=\begin{pmatrix}1&0\\ 0&1\end{pmatrix},
  321. ( - ( 1 ) α - 1 + α 6 x α 3 + α 1 x - ( α - 7 + α 7 x + α 7 x 2 ) ) ( S ( x ) Γ ( x ) x 6 ) = ( α 4 + α 7 x + α 5 x 2 + α 3 x 3 + α 1 x 4 + α - 1 x 5 α 7 + α 0 x ) . \begin{pmatrix}-(1)&\alpha^{-1}+\alpha^{6}x\\ \alpha^{3}+\alpha^{1}x&-(\alpha^{-7}+\alpha^{7}x+\alpha^{7}x^{2})\end{pmatrix}% \begin{pmatrix}S(x)\Gamma(x)\\ x^{6}\end{pmatrix}=\begin{pmatrix}\alpha^{4}+\alpha^{7}x+\alpha^{5}x^{2}+% \alpha^{3}x^{3}+\alpha^{1}x^{4}+\alpha^{-1}x^{5}\\ \alpha^{7}+\alpha^{0}x\end{pmatrix}.
  322. S ( x ) Γ ( x ) ( α 3 + α 1 x ) - ( α - 7 + α 7 x + α 7 x 2 ) x 6 = α 7 + α 0 x . S(x)\Gamma(x)(\alpha^{3}+\alpha^{1}x)-(\alpha^{-7}+\alpha^{7}x+\alpha^{7}x^{2}% )x^{6}=\alpha^{7}+\alpha^{0}x.
  323. Λ ( x ) = α 3 + α 1 x . \Lambda(x)=\alpha^{3}+\alpha^{1}x.
  324. λ 0 1. \lambda_{0}\neq 1.
  325. Λ ( x ) \Lambda(x)
  326. α 3 - 1 . \alpha^{3-1}.
  327. Ξ ( x ) = Γ ( x ) Λ ( x ) = α 3 + α - 7 x + α - 4 x 2 + α 5 x 3 , \Xi(x)=\Gamma(x)\Lambda(x)=\alpha^{3}+\alpha^{-7}x+\alpha^{-4}x^{2}+\alpha^{5}% x^{3},
  328. Ω ( x ) = S ( x ) Ξ ( x ) mod x 6 = α 7 + α 0 x . \Omega(x)=S(x)\Xi(x)\bmod x^{6}=\alpha^{7}+\alpha^{0}x.
  329. e j = - Ω ( α - i j ) / Ξ ( α - i j ) , e_{j}=-\Omega(\alpha^{-i_{j}})/\Xi^{\prime}(\alpha^{-i_{j}}),
  330. α - i j \alpha^{-i_{j}}
  331. Ξ ( x ) . \Xi(x).
  332. Ξ ( x ) = α - 7 + α 5 x 2 . \Xi^{\prime}(x)=\alpha^{-7}+\alpha^{5}x^{2}.
  333. e 1 = - Ω ( α 4 ) / Ξ ( α 4 ) = ( α 7 + α 4 ) / ( α - 7 + α - 2 ) = α 3 / α 3 = 1 , e_{1}=-\Omega(\alpha^{4})/\Xi^{\prime}(\alpha^{4})=(\alpha^{7}+\alpha^{4})/(% \alpha^{-7}+\alpha^{-2})=\alpha^{3}/\alpha^{3}=1,
  334. e 2 = - Ω ( α 7 ) / Ξ ( α 7 ) = ( α 7 + α 7 ) / ( α - 7 + α 4 ) = 0 / α 5 = 0 , e_{2}=-\Omega(\alpha^{7})/\Xi^{\prime}(\alpha^{7})=(\alpha^{7}+\alpha^{7})/(% \alpha^{-7}+\alpha^{4})=0/\alpha^{5}=0,
  335. e 3 = - Ω ( α 2 ) / Ξ ( α 2 ) = ( α 7 + α 2 ) / ( α - 7 + α - 6 ) = α - 3 / α - 3 = 1. e_{3}=-\Omega(\alpha^{2})/\Xi^{\prime}(\alpha^{2})=(\alpha^{7}+\alpha^{2})/(% \alpha^{-7}+\alpha^{-6})=\alpha^{-3}/\alpha^{-3}=1.
  336. e 3 = 1 e_{3}=1

BCS_theory.html

  1. Δ ( T = 0 ) = 1.764 k B T c , \Delta(T=0)=1.764\,k_{B}T_{c},
  2. Δ ( T T c ) 3.07 k B T c 1 - ( T / T c ) \Delta(T\to T_{c})\approx 3.07\,k_{B}T_{c}\sqrt{1-(T/T_{c})}
  3. k B T c = 1.13 E D e - 1 / N ( 0 ) V , k_{B}\,T_{c}=1.13E_{D}\,{e^{-1/N(0)\,V}},

Beam_diameter.html

  1. 2 w = 2 FWHM ln 2 = 1.699 × FWHM 2w=\frac{\sqrt{2}\ \mathrm{FWHM}}{\sqrt{\ln 2}}=1.699\times\mathrm{FWHM}
  2. 2 w 2w
  3. I ( x , y ) I(x,y)
  4. D 4 σ = 4 σ = 4 - - I ( x , y ) ( x - x ¯ ) 2 d x d y - - I ( x , y ) d x d y D4\sigma=4\sigma=4\sqrt{\frac{\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}I(% x,y)(x-\bar{x})^{2}\,dx\,dy}{\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}I(x% ,y)\,dx\,dy}}
  5. x ¯ = - - I ( x , y ) x d x d y - - I ( x , y ) d x d y \bar{x}=\frac{\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}I(x,y)x\,dx\,dy}{% \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}I(x,y)\,dx\,dy}
  6. d σ x = 2 2 ( x 2 + y 2 + γ ( ( x 2 - y 2 ) 2 + 4 x y 2 ) 1 / 2 ) 1 / 2 d_{\sigma x}=2\sqrt{2}\left(\langle x^{2}\rangle+\langle y^{2}\rangle+\gamma% \left(\left(\langle x^{2}\rangle-\langle y^{2}\rangle\right)^{2}+4\langle xy% \rangle^{2}\right)^{1/2}\right)^{1/2}
  7. d σ y = 2 2 ( x 2 + y 2 - γ ( ( x 2 - y 2 ) 2 + 4 x y 2 ) 1 / 2 ) 1 / 2 . d_{\sigma y}=2\sqrt{2}\left(\langle x^{2}\rangle+\langle y^{2}\rangle-\gamma% \left(\left(\langle x^{2}\rangle-\langle y^{2}\rangle\right)^{2}+4\langle xy% \rangle^{2}\right)^{1/2}\right)^{1/2}.
  8. x y \langle xy\rangle
  9. x = 1 P I ( x , y ) x d x d y , \langle x\rangle=\frac{1}{P}\int{I(x,y)xdxdy},
  10. y = 1 P I ( x , y ) y d x d y \langle y\rangle=\frac{1}{P}\int{I(x,y)ydxdy}
  11. x 2 = 1 P I ( x , y ) ( x - x ) 2 d x d y , \langle x^{2}\rangle=\frac{1}{P}\int{I(x,y)(x-\langle x\rangle)^{2}dxdy},
  12. x y = 1 P I ( x , y ) ( x - x ) ( y - y ) d x d y , \langle xy\rangle=\frac{1}{P}\int{I(x,y)(x-\langle x\rangle)(y-\langle y% \rangle)dxdy},
  13. y 2 = 1 P I ( x , y ) ( y - y ) 2 d x d y , \langle y^{2}\rangle=\frac{1}{P}\int{I(x,y)(y-\langle y\rangle)^{2}dxdy},
  14. P = I ( x , y ) d x d y P=\int{I(x,y)dxdy}
  15. γ = sgn ( x 2 - y 2 ) = x 2 - y 2 | x 2 - y 2 | . \gamma=\operatorname{sgn}\left(\langle x^{2}\rangle-\langle y^{2}\rangle\right% )=\frac{\langle x^{2}\rangle-\langle y^{2}\rangle}{|\langle x^{2}\rangle-% \langle y^{2}\rangle|}.
  16. ϕ \phi
  17. x x
  18. y y
  19. ϕ = 1 2 arctan 2 x y x 2 - y 2 . \phi=\frac{1}{2}\arctan\frac{2\langle xy\rangle}{\langle x^{2}\rangle-\langle y% ^{2}\rangle}.

Beam_divergence.html

  1. Θ \Theta
  2. Θ = 2 arctan ( D f - D i 2 l ) . \Theta=2\arctan\left(\frac{D_{f}-D_{i}}{2l}\right).
  3. D m D_{m}
  4. Θ = D m f , \Theta=\frac{D_{m}}{f},\,
  5. θ = Θ / 2 \theta=\Theta/2
  6. θ = λ π w , \theta={\lambda\over\pi w},
  7. λ \lambda
  8. w w

Beef.html

  1. YG = 2.5 + ( 2.5 × adjusted fat thickness ) + ( 0.2 × percent KPH ) + ( 0.0038 × HCW ) - ( 0.32 × REA ) {\,\text{YG}=2.5+\left(2.5\times\,\text{adjusted fat thickness}\right)+\left(0% .2\times\,\text{percent KPH}\right)+\left(0.0038\times\,\text{HCW}\right)-% \left(0.32\times\,\text{REA}\right)}

Beer–Lambert_law.html

  1. T = Φ e t Φ e i = e - τ = 10 - A , T=\frac{\Phi_{\mathrm{e}}^{\mathrm{t}}}{\Phi_{\mathrm{e}}^{\mathrm{i}}}=e^{-% \tau}=10^{-A},
  2. T = e - i = 1 N σ i 0 n i ( z ) d z = 10 - i = 1 N ε i 0 c i ( z ) d z , T=e^{-\sum_{i=1}^{N}\sigma_{i}\int_{0}^{\ell}n_{i}(z)\mathrm{d}z}=10^{-\sum_{i% =1}^{N}\varepsilon_{i}\int_{0}^{\ell}c_{i}(z)\mathrm{d}z},
  3. τ = i = 1 N τ i = i = 1 N σ i 0 n i ( z ) d z , \tau=\sum_{i=1}^{N}\tau_{i}=\sum_{i=1}^{N}\sigma_{i}\int_{0}^{\ell}n_{i}(z)\,% \mathrm{d}z,
  4. A = i = 1 N A i = i = 1 N ε i 0 c i ( z ) d z , A=\sum_{i=1}^{N}A_{i}=\sum_{i=1}^{N}\varepsilon_{i}\int_{0}^{\ell}c_{i}(z)\,% \mathrm{d}z,
  5. ε i = N A ln 10 σ i , \varepsilon_{i}=\frac{\mathrm{N_{A}}}{\ln{10}}\,\sigma_{i},
  6. c i = n i N A , c_{i}=\frac{n_{i}}{\mathrm{N_{A}}},
  7. T = e - i = 1 N σ i n i = 10 - i = 1 N ε i c i , T=e^{-\sum_{i=1}^{N}\sigma_{i}n_{i}\ell}=10^{-\sum_{i=1}^{N}\varepsilon_{i}c_{% i}\ell},
  8. τ = i = 1 N σ i n i , \tau=\sum_{i=1}^{N}\sigma_{i}n_{i}\ell,
  9. A = i = 1 N ε i c i . A=\sum_{i=1}^{N}\varepsilon_{i}c_{i}\ell.
  10. μ ( z ) = i = 1 N μ i ( z ) = i = 1 N σ i n i ( z ) , \mu(z)=\sum_{i=1}^{N}\mu_{i}(z)=\sum_{i=1}^{N}\sigma_{i}n_{i}(z),
  11. μ 10 ( z ) = i = 1 N μ 10 , i ( z ) = i = 1 N ε i c i ( z ) \mu_{10}(z)=\sum_{i=1}^{N}\mu_{10,i}(z)=\sum_{i=1}^{N}\varepsilon_{i}c_{i}(z)
  12. T = e - 0 μ ( z ) d z = 10 - 0 μ 10 ( z ) d z , T=e^{-\int_{0}^{\ell}\mu(z)\mathrm{d}z}=10^{-\int_{0}^{\ell}\mu_{10}(z)\mathrm% {d}z},
  13. τ = 0 μ ( z ) d z , \tau=\int_{0}^{\ell}\mu(z)\,\mathrm{d}z,
  14. A = 0 μ 10 ( z ) d z . A=\int_{0}^{\ell}\mu_{10}(z)\,\mathrm{d}z.
  15. T = e - μ = 10 - μ 10 , T=e^{-\mu\ell}=10^{-\mu_{10}\ell},
  16. τ = μ , \tau=\mu\ell,
  17. A = μ 10 . A=\mu_{10}\ell.
  18. d Φ e d z ( z ) = - μ ( z ) Φ e ( z ) . \frac{\mathrm{d}\Phi_{\mathrm{e}}}{\mathrm{d}z}(z)=-\mu(z)\Phi_{\mathrm{e}}(z).
  19. e 0 z μ ( z ) d z e^{\int_{0}^{z}\mu(z^{\prime})\mathrm{d}z^{\prime}}
  20. d Φ e d z ( z ) e 0 z μ ( z ) d z + μ ( z ) Φ e ( z ) e 0 z μ ( z ) d z = 0 , \frac{\mathrm{d}\Phi_{\mathrm{e}}}{\mathrm{d}z}(z)\,e^{\int_{0}^{z}\mu(z^{% \prime})\mathrm{d}z^{\prime}}+\mu(z)\Phi_{\mathrm{e}}(z)\,e^{\int_{0}^{z}\mu(z% ^{\prime})\mathrm{d}z^{\prime}}=0,
  21. d d z ( Φ e ( z ) e 0 z μ ( z ) d z ) = 0. \frac{\mathrm{d}}{\mathrm{d}z}\bigl(\Phi_{\mathrm{e}}(z)\,e^{\int_{0}^{z}\mu(z% ^{\prime})\mathrm{d}z^{\prime}}\bigr)=0.
  22. Φ e t = Φ e i e - 0 μ ( z ) d z , \Phi_{\mathrm{e}}^{\mathrm{t}}=\Phi_{\mathrm{e}}^{\mathrm{i}}\,e^{-\int_{0}^{% \ell}\mu(z)\mathrm{d}z},
  23. T = Φ e t Φ e i = e - 0 μ ( z ) d z . T=\frac{\Phi_{\mathrm{e}}^{\mathrm{t}}}{\Phi_{\mathrm{e}}^{\mathrm{i}}}=e^{-% \int_{0}^{\ell}\mu(z)\mathrm{d}z}.
  24. T = e - 0 ln 10 μ 10 ( z ) d z = ( e - 0 μ 10 ( z ) d z ) ln 10 = 10 - 0 μ 10 ( z ) d z . T=e^{-\int_{0}^{\ell}\ln{10}\,\mu_{10}(z)\mathrm{d}z}=\bigl(e^{-\int_{0}^{\ell% }\mu_{10}(z)\mathrm{d}z}\bigr)^{\ln{10}}=10^{-\int_{0}^{\ell}\mu_{10}(z)% \mathrm{d}z}.
  25. T = e - i = 1 N σ i 0 n i ( z ) d z . T=e^{-\sum_{i=1}^{N}\sigma_{i}\int_{0}^{\ell}n_{i}(z)\mathrm{d}z}.
  26. T = e - i = 1 N ln 10 N A ε i 0 n i ( z ) d z = ( e - i = 1 N ε i 0 n i ( z ) N A d z ) ln 10 = 10 - i = 1 N ε i 0 c i ( z ) d z . T=e^{-\sum_{i=1}^{N}\frac{\ln{10}}{\mathrm{N_{A}}}\varepsilon_{i}\int_{0}^{% \ell}n_{i}(z)\mathrm{d}z}=\Bigl(e^{-\sum_{i=1}^{N}\varepsilon_{i}\int_{0}^{% \ell}\frac{n_{i}(z)}{\mathrm{N_{A}}}\mathrm{d}z}\Bigr)^{\ln{10}}=10^{-\sum_{i=% 1}^{N}\varepsilon_{i}\int_{0}^{\ell}c_{i}(z)\mathrm{d}z}.
  27. c = μ 10 ( λ ) ε ( λ ) . c=\frac{\mu_{10}(\lambda)}{\varepsilon(\lambda)}.
  28. μ 10 ( λ ) = ε 1 ( λ ) c 1 + ε 2 ( λ ) c 2 . \mu_{10}(\lambda)=\varepsilon_{1}(\lambda)c_{1}+\varepsilon_{2}(\lambda)c_{2}.
  29. T = e - m ( τ a + τ g + τ RS + τ NO 2 + τ w + τ O 3 + τ r + ) , T=e^{-m(\tau_{\mathrm{a}}+\tau_{\mathrm{g}}+\tau_{\mathrm{RS}}+\tau_{\mathrm{% NO_{2}}}+\tau_{\mathrm{w}}+\tau_{\mathrm{O_{3}}}+\tau_{\mathrm{r}}+\ldots)},

Bell's_theorem.html

  1. ± 0.5 ±0.5
  2. ± 1 2 = ± 0.70... , \pm\tfrac{1}{\sqrt{2}}=\pm 0.70...,
  3. ± 0.5 ±0.5
  4. ρ ( a , c ) - ρ ( b , a ) - ρ ( b , c ) 1 , \rho(a,c)-\rho(b,a)-\rho(b,c)\leq 1,
  5. ρ ρ
  6. a , b a,b
  7. X , Y X,Y
  8. θ θ
  9. f = θ 2 / 2 f=\theta^{2}/2
  10. 2 2 = 4 2^{2}=4
  11. ( 1 ) ρ ( a , b ) + ρ ( a , b ) + ρ ( a , b ) - ρ ( a , b ) 2 (1)\quad\rho(a,b)+\rho(a,b^{\prime})+\rho(a^{\prime},b)-\rho(a^{\prime},b^{% \prime})\leq 2
  12. ρ ρ
  13. a = a + π a^{\prime}=a+\pi
  14. b = c b^{\prime}=c
  15. ρ ( a , a + π ) = 1 \rho(a,a+\pi)=1
  16. ρ ( b , a + π ) = - ρ ( b , a ) \rho(b,a+\pi)=-\rho(b,a)
  17. Λ Λ
  18. λ Λ λ∈Λ
  19. a a
  20. A ( a , λ ) \scriptstyle A(a,\lambda)
  21. b b
  22. B ( b , λ ) \scriptstyle B(b,\lambda)
  23. Λ Λ
  24. ρ ρ
  25. X X
  26. Λ Λ
  27. ρ ρ
  28. E ( X ) = Λ X ( λ ) ρ ( λ ) d λ \operatorname{E}(X)=\int_{\Lambda}X(\lambda)\rho(\lambda)d\lambda
  29. ρ ρ
  30. A = A ( a , λ ) , A = A ( a , λ ) , B = B ( b , λ ) , B = B ( b , λ ) A=A(a,\lambda),A^{\prime}=A(a^{\prime},\lambda),B=B(b,\lambda),B^{\prime}=B(b^% {\prime},\lambda)
  31. ± 1 ±1
  32. λ λ
  33. λ Λ λ∈Λ
  34. B + B B+B^{\prime}
  35. B - B B-B^{\prime}
  36. ± 2 ±2
  37. A B + A B + A B - A B = A ( B + B ) + A ( B - B ) 2 AB+AB^{\prime}+A^{\prime}B-A^{\prime}B^{\prime}=A(B+B^{\prime})+A^{\prime}(B-B% ^{\prime})\leq 2
  38. ρ ( a , b ) + ρ ( a , b ) + ρ ( a , b ) - ρ ( a , b ) = Λ A B ρ + Λ A B ρ + Λ A B ρ - Λ A B ρ = Λ ( A B + A B + A B - A B ) ρ = Λ ( A ( B + B ) + A ( B - B ) ) ρ 2 \begin{aligned}\displaystyle\rho(a,b)+\rho(a,b^{\prime})+\rho(a^{\prime},b)-% \rho(a^{\prime},b^{\prime})&\displaystyle=\int_{\Lambda}AB\rho+\int_{\Lambda}% AB^{\prime}\rho+\int_{\Lambda}A^{\prime}B\rho-\int_{\Lambda}A^{\prime}B^{% \prime}\rho\\ &\displaystyle=\int_{\Lambda}(AB+AB^{\prime}+A^{\prime}B-A^{\prime}B^{\prime})% \rho\\ &\displaystyle=\int_{\Lambda}(A(B+B^{\prime})+A^{\prime}(B-B^{\prime}))\rho\\ &\displaystyle\leq 2\end{aligned}
  39. A , A , B , B A,A^{\prime},B,B^{\prime}
  40. ± 1 ±1
  41. A B + A B + A B - A B = A ( B + B ) + A ( B - B ) 2. AB+AB^{\prime}+A^{\prime}B-A^{\prime}B^{\prime}=A(B+B^{\prime})+A^{\prime}(B-B% ^{\prime})\leq 2.
  42. x z x′−z′
  43. x z x−z
  44. S x = [ 0 1 1 0 ] , S z = [ 1 0 0 - 1 ] S_{x}=\begin{bmatrix}0&1\\ 1&0\end{bmatrix},\quad S_{z}=\begin{bmatrix}1&0\\ 0&-1\end{bmatrix}
  45. ± 1 ±1
  46. | + x , | - x . \left|+x\right\rangle,\quad|-x\rangle.
  47. ϕ \phi
  48. | ϕ = 1 2 ( | + x | - x - | - x | + x ) |\phi\rangle=\frac{1}{\sqrt{2}}\left(|+x\rangle\otimes|-x\rangle-|-x\rangle% \otimes|+x\rangle\right)
  49. A ( a ) = S z I A ( a ) = S x I B ( b ) = - 1 2 I ( S z + S x ) B ( b ) = 1 2 I ( S z - S x ) \begin{aligned}\displaystyle A(a)&\displaystyle=S_{z}\otimes I\\ \displaystyle A(a^{\prime})&\displaystyle=S_{x}\otimes I\\ \displaystyle B(b)&\displaystyle=-\frac{1}{\sqrt{2}}\ I\otimes(S_{z}+S_{x})\\ \displaystyle B(b^{\prime})&\displaystyle=\frac{1}{\sqrt{2}}\ I\otimes(S_{z}-S% _{x})\end{aligned}
  50. B ( b ) , B ( b ) \scriptstyle B(b^{\prime}),B(b)
  51. A A
  52. B B
  53. A ( a ) B ( b ) = A ( a ) B ( b ) = A ( a ) B ( b ) = 1 2 \langle A(a)B(b)\rangle=\langle A(a^{\prime})B(b)\rangle=\langle A(a^{\prime})% B(b^{\prime})\rangle=\tfrac{1}{\sqrt{2}}
  54. A ( a ) B ( b ) = - 1 2 \langle A(a)B(b^{\prime})\rangle=-\tfrac{1}{\sqrt{2}}
  55. A ( a ) B ( b ) + A ( a ) B ( b ) + A ( a ) B ( b ) - A ( a ) B ( b ) = 4 2 = 2 2 > 2 \langle A(a)B(b)\rangle+\langle A(a^{\prime})B(b^{\prime})\rangle+\langle A(a^% {\prime})B(b)\rangle-\langle A(a)B(b^{\prime})\rangle=\tfrac{4}{\sqrt{2}}=2% \sqrt{2}>2
  56. 2 2 2\sqrt{2}
  57. 1 2 ( | V | V + | H | H ) \tfrac{1}{\sqrt{2}}\left(|V\rangle\otimes|V\rangle+|H\rangle\otimes|H\rangle\right)
  58. | V |V\rangle
  59. | H |H\rangle
  60. θ θ
  61. c o s ( 2 θ ) cos(2θ)
  62. π π
  63. 2 π

Benford's_law.html

  1. P ( d ) = log 10 ( d + 1 ) - log 10 ( d ) = log 10 ( d + 1 d ) = log 10 ( 1 + 1 d ) . P(d)=\log_{10}(d+1)-\log_{10}(d)=\log_{10}\left(\frac{d+1}{d}\right)=\log_{10}% \left(1+\frac{1}{d}\right).
  2. m = N max i = 1 9 { Pr ( X has FSD = i ) - log 10 ( 1 + 1 / i ) } , m=\sqrt{N}\cdot\operatorname*{max}_{i=1}^{9}\Big\{\Pr(X\,\text{ has FSD}=i)-% \log_{10}(1+1/i)\Big\},
  3. d = N i = 1 9 [ Pr ( X has FSD = i ) - log 10 ( 1 + 1 / i ) ] 2 , d=\sqrt{N\cdot\sum_{i=1}^{9}\Big[\Pr(X\,\text{ has FSD}=i)-\log_{10}(1+1/i)% \Big]^{2}},
  4. N N
  5. z = | p o - p e | - 1 2 n s i z=\frac{\,|p_{o}-p_{e}|-\frac{1}{2n}\,}{s_{i}}
  6. s i = [ p e ( 1 - p e ) n ] 1 / 2 , s_{i}=\left[\frac{p_{e}(1-p_{e})}{n}\right]^{1/2},
  7. log 10 ( n + 1 ) - log 10 ( n ) = log 10 ( 1 + 1 n ) \log_{10}\left(n+1\right)-\log_{10}\left(n\right)=\log_{10}\left(1+\frac{1}{n}\right)
  8. log 10 ( 1 + 1 12 ) + log 10 ( 1 + 1 22 ) + + log 10 ( 1 + 1 92 ) 0.109 \log_{10}\left(1+\frac{1}{12}\right)+\log_{10}\left(1+\frac{1}{22}\right)+% \cdots+\log_{10}\left(1+\frac{1}{92}\right)\approx 0.109
  9. k = 10 n - 2 10 n - 1 - 1 log 10 ( 1 + 1 10 k + d ) \sum_{k=10^{n-2}}^{10^{n-1}-1}\log_{10}\left(1+\frac{1}{10k+d}\right)
  10. P ( log x ) d ( log x ) = ( 1 / x ) P ( log x ) d x P(\log x)d(\log x)=(1/x)P(\log x)dx

Berkelium.html

  1. × 10 1 0 \times 10^{1}0
  2. 3 ¯ \overline{3}
  3. 3 ¯ \overline{3}
  4. × 10 - 3 \times 10^{-}3
  5. Am 95 241 + 2 4 He 97 243 Bk + 2 0 1 n \mathrm{{}^{241}_{\ 95}Am\ +\ ^{4}_{2}He\ \longrightarrow\ ^{243}_{\ 97}Bk\ +% \ 2\ ^{1}_{0}n}
  6. U 92 238 ( n , γ ) 92 239 U β - 23.5 min 93 239 Np β - 2.3565 d 94 239 Pu \mathrm{{}^{238}_{\ 92}U\ \xrightarrow{(n,\gamma)}\ ^{239}_{\ 92}U\ % \xrightarrow[23.5\ min]{\beta^{-}}\ ^{239}_{\ 93}Np\ \xrightarrow[2.3565\ d]{% \beta^{-}}\ ^{239}_{\ 94}Pu}
  7. Pu 94 239 4 ( n , γ ) 94 243 Pu β - 4.956 h 95 243 Am ( n , γ ) 95 244 Am β - 10.1 h 96 244 Cm ; Cm 96 244 5 ( n , γ ) 96 249 Cm \mathrm{{}^{239}_{\ 94}Pu\ \xrightarrow{4(n,\gamma)}\ ^{243}_{\ 94}Pu\ % \xrightarrow[4.956\ h]{\beta^{-}}\ ^{243}_{\ 95}Am\ \xrightarrow{(n,\gamma)}\ % ^{244}_{\ 95}Am\ \xrightarrow[10.1\ h]{\beta^{-}}\ ^{244}_{\ 96}Cm}\quad;\quad% \mathrm{{}^{244}_{\ 96}Cm\ \xrightarrow{5(n,\gamma)}\ ^{249}_{\ 96}Cm}
  8. Cm 96 249 β - 64.15 min 97 249 Bk β - 330 d 98 249 Cf \mathrm{{}^{249}_{\ 96}Cm\ \xrightarrow[64.15\ min]{\beta^{-}}\ ^{249}_{\ 97}% Bk\ \xrightarrow[330\ d]{\beta^{-}}\ ^{249}_{\ 98}Cf}
  9. Bk 97 249 ( n , γ ) 97 250 Bk β - 3.212 h 98 250 Cf \mathrm{{}^{249}_{\ 97}Bk\ \xrightarrow{(n,\gamma)}\ ^{250}_{\ 97}Bk\ % \xrightarrow[3.212\ h]{\beta^{-}}\ ^{250}_{\ 98}Cf}
  10. Cm 96 244 ( α , n ) 98 247 Cf 3.11 h ϵ 97 247 Bk \mathrm{{}^{244}_{\ 96}Cm\ \xrightarrow[]{(\alpha,n)}\ ^{247}_{\ 98}Cf\ % \xrightarrow[3.11\ h]{\epsilon}\ ^{247}_{\ 97}Bk}
  11. Cm 96 244 ( α , p ) 97 247 Bk \mathrm{{}^{244}_{\ 96}Cm\ \xrightarrow[]{(\alpha,p)}\ ^{247}_{\ 97}Bk}
  12. U 92 235 + 5 11 B 97 242 Bk + 4 0 1 n ; 90 232 Th + 7 14 N 97 242 Bk + 4 0 1 n \mathrm{{}^{235}_{\ 92}U\ +\ ^{11}_{\ 5}B\ \longrightarrow\ ^{242}_{\ 97}Bk\ +% \ 4\ ^{1}_{0}n\quad;\quad^{232}_{\ 90}Th\ +\ ^{14}_{\ 7}N\ \longrightarrow\ ^{% 242}_{\ 97}Bk\ +\ 4\ ^{1}_{0}n}
  13. U 92 238 + 5 10 B 97 242 Bk + 6 0 1 n ; 90 232 Th + 7 15 N 97 242 Bk + 5 0 1 n \mathrm{{}^{238}_{\ 92}U\ +\ ^{10}_{\ 5}B\ \longrightarrow\ ^{242}_{\ 97}Bk\ +% \ 6\ ^{1}_{0}n\quad;\quad^{232}_{\ 90}Th\ +\ ^{15}_{\ 7}N\ \longrightarrow\ ^{% 242}_{\ 97}Bk\ +\ 5\ ^{1}_{0}n}
  14. BkF 3 + 3 Li Bk + 3 LiF \mathrm{BkF_{3}\ +\ 3\ Li\ \longrightarrow\ Bk\ +\ 3\ LiF}
  15. 2 BkO 2 + H 2 Bk 2 O 3 + H 2 O \mathrm{2\ BkO_{2}\ +\ H_{2}\ \longrightarrow\ Bk_{2}O_{3}\ +\ H_{2}O}

Bernhard_Riemann.html

  1. g = w / 2 - n + 1 g=w/2-n+1
  2. n n
  3. w w
  4. g > 1 g>1
  5. ( 3 g - 3 ) (3g-3)
  6. \mathbb{C}
  7. C n / Ω C^{n}/\Omega
  8. Ω \Omega
  9. n n
  10. π ( x ) \pi(x)
  11. L i ( x ) Li(x)
  12. π ( x ) \pi(x)

Bernoulli's_inequality.html

  1. ( 1 + x ) r 1 + r x (1+x)^{r}\geq 1+rx\!
  2. ( 1 + x ) r > 1 + r x (1+x)^{r}>1+rx\!
  3. ( 1 + x ) 0 1 + 0 x (1+x)^{0}\geq 1+0x\,
  4. ( 1 + x ) k 1 + k x . (1+x)^{k}\geq 1+kx.\,
  5. ( 1 + x ) ( 1 + x ) k ( 1 + x ) ( 1 + k x ) (by hypothesis, since ( 1 + x ) 0 ) ( 1 + x ) k + 1 1 + k x + x + k x 2 , ( 1 + x ) k + 1 1 + ( k + 1 ) x + k x 2 . \begin{aligned}&\displaystyle{}\qquad(1+x)(1+x)^{k}\geq(1+x)(1+kx)\quad\,\text% {(by hypothesis, since }(1+x)\geq 0)\\ &\displaystyle\iff(1+x)^{k+1}\geq 1+kx+x+kx^{2},\\ &\displaystyle\iff(1+x)^{k+1}\geq 1+(k+1)x+kx^{2}.\end{aligned}
  6. ( 1 + x ) r 1 + r x (1+x)^{r}\geq 1+rx\!
  7. ( 1 + x ) r 1 + r x (1+x)^{r}\leq 1+rx\!
  8. ( 1 + x ) r e r x , (1+x)^{r}\leq e^{rx},\!
  9. 0 x 1 0\leq x\leq 1
  10. ( 1 - x ) t 1 - x t . (1-x)^{t}\geq 1-xt.
  11. t = 1 + 1 + + 1 1 + y + y 2 + + y t - 1 = 1 - y t 1 - y t=1+1+\dots+1\geq 1+y+y^{2}+\ldots+y^{t-1}=\frac{1-y^{t}}{1-y}
  12. x t 1 - ( 1 - x ) t . xt\geq 1-(1-x)^{t}.
  13. t = a b 1 t=\frac{a}{b}\leq 1
  14. b b
  15. ( 1 + x ) (1+x)
  16. a a
  17. 1 , 1 , , ( 1 + x ) , ( 1 + x ) , ( 1 + x ) 1,1,\ldots,(1+x),(1+x),\ldots(1+x)
  18. ( 1 + x ) a / b ( 1 + a b x ) (1+x)^{a/b}\leq\left(1+\frac{a}{b}x\right)
  19. ( 1 + x ) t ( 1 + t x ) (1+x)^{t}\leq\left(1+tx\right)
  20. t 1 t\leq 1
  21. s 1 s\geq 1
  22. z = a b x z=\frac{a}{b}x
  23. x - 1 , z - a b x\geq-1,z\geq-\frac{a}{b}
  24. s = 1 t = b a 1 s=\frac{1}{t}=\frac{b}{a}\geq 1
  25. ( 1 + z ) s 1 + s z \left(1+z\right)^{s}\geq 1+sz
  26. s 1 s\geq 1
  27. t 1 t\leq 1
  28. s 1 s\geq 1

Bernoulli's_principle.html

  1. v 2 2 + g z + p ρ = constant {v^{2}\over 2}+gz+{p\over\rho}=\,\text{constant}
  2. v v\,
  3. g g\,
  4. z z\,
  5. p p\,
  6. ρ \rho\,
  7. v 2 2 + Ψ + p ρ = constant {v^{2}\over 2}+\Psi+{p\over\rho}=\,\text{constant}
  8. ρ \rho
  9. 1 2 ρ v 2 + ρ g z + p = constant \tfrac{1}{2}\,\rho\,v^{2}\,+\,\rho\,g\,z\,+\,p\,=\,\,\text{constant}\,
  10. q + ρ g h = p 0 + ρ g z = constant q\,+\,\rho\,g\,h\,=\,p_{0}\,+\,\rho\,g\,z\,=\,\,\text{constant}\,
  11. q = 1 2 ρ v 2 q\,=\,\tfrac{1}{2}\,\rho\,v^{2}
  12. h = z + p ρ g h\,=\,z\,+\,\frac{p}{\rho g}
  13. p 0 = p + q p_{0}\,=\,p\,+\,q\,
  14. H = z + p ρ g + v 2 2 g = h + v 2 2 g , H\,=\,z\,+\,\frac{p}{\rho g}\,+\,\frac{v^{2}}{2\,g}\,=\,h\,+\,\frac{v^{2}}{2\,% g},
  15. p + q = p 0 p+q=p_{0}\,
  16. φ t + 1 2 v 2 + p ρ + g z = f ( t ) , \frac{\partial\varphi}{\partial t}+\tfrac{1}{2}v^{2}+\frac{p}{\rho}+gz=f(t),
  17. Φ = φ - t 0 t f ( τ ) d τ , \Phi=\varphi-\int_{t_{0}}^{t}f(\tau)\,\,\text{d}\tau,
  18. Φ t + 1 2 v 2 + p ρ + g z = 0. \displaystyle\frac{\partial\Phi}{\partial t}+\tfrac{1}{2}v^{2}+\frac{p}{\rho}+% gz=0.
  19. v 2 2 + p 1 p d p ~ ρ ( p ~ ) + Ψ = constant \frac{v^{2}}{2}+\int_{p_{1}}^{p}\frac{d\tilde{p}}{\rho(\tilde{p})}\ +\Psi=\,% \text{constant}
  20. v 2 2 + g z + ( γ γ - 1 ) p ρ = constant \frac{v^{2}}{2}+gz+\left(\frac{\gamma}{\gamma-1}\right)\frac{p}{\rho}=\,\text{constant}
  21. v 2 2 + ( γ γ - 1 ) p ρ = ( γ γ - 1 ) p 0 ρ 0 \frac{v^{2}}{2}+\left(\frac{\gamma}{\gamma-1}\right)\frac{p}{\rho}=\left(\frac% {\gamma}{\gamma-1}\right)\frac{p_{0}}{\rho_{0}}
  22. v 2 2 + Ψ + w = constant {v^{2}\over 2}+\Psi+w=\,\text{constant}
  23. w = ϵ + p ρ w=\epsilon+\frac{p}{\rho}
  24. v 2 2 + w = w 0 {v^{2}\over 2}+w=w_{0}
  25. m d v d t = F m\frac{\operatorname{d}v}{\operatorname{d}t}=F
  26. ρ A d x d v d t = - A d p \rho A\operatorname{d}x\frac{\operatorname{d}v}{\operatorname{d}t}=-A% \operatorname{d}p
  27. ρ d v d t = - d p d x \rho\frac{\operatorname{d}v}{\operatorname{d}t}=-\frac{\operatorname{d}p}{% \operatorname{d}x}
  28. d v d t = d v d x d x d t = d v d x v = d d x ( v 2 2 ) . \frac{\operatorname{d}v}{\operatorname{d}t}=\frac{\operatorname{d}v}{% \operatorname{d}x}\frac{\operatorname{d}x}{\operatorname{d}t}=\frac{% \operatorname{d}v}{\operatorname{d}x}v=\frac{d}{\operatorname{d}x}\left(\frac{% v^{2}}{2}\right).
  29. d d x ( ρ v 2 2 + p ) = 0 \frac{\operatorname{d}}{\operatorname{d}x}\left(\rho\frac{v^{2}}{2}+p\right)=0
  30. v 2 2 + p ρ = C \frac{v^{2}}{2}+\frac{p}{\rho}=C
  31. W = Δ E kin . W=\Delta E\text{kin}.\;
  32. ρ A 1 s 1 \displaystyle\rho A_{1}s_{1}
  33. W pressure = F 1 , pressure s 1 - F 2 , pressure s 2 = p 1 A 1 s 1 - p 2 A 2 s 2 = Δ m p 1 ρ - Δ m p 2 ρ . W\text{pressure}=F_{1,\,\text{pressure}}\;s_{1}\,-\,F_{2,\,\text{pressure}}\;s% _{2}=p_{1}A_{1}s_{1}-p_{2}A_{2}s_{2}=\Delta m\,\frac{p_{1}}{\rho}-\Delta m\,% \frac{p_{2}}{\rho}.\;
  34. Δ E pot,gravity = Δ m g z 2 - Δ m g z 1 . \Delta E\text{pot,gravity}=\Delta m\,gz_{2}-\Delta m\,gz_{1}.\;
  35. W gravity = - Δ E pot,gravity = Δ m g z 1 - Δ m g z 2 . W\text{gravity}=-\Delta E\text{pot,gravity}=\Delta m\,gz_{1}-\Delta m\,gz_{2}.\;
  36. Δ t \Delta t
  37. W = W pressure + W gravity . W=W\text{pressure}+W\text{gravity}.\,
  38. Δ E kin = 1 2 Δ m v 2 2 - 1 2 Δ m v 1 2 . \Delta E\text{kin}=\frac{1}{2}\Delta m\,v_{2}^{2}-\frac{1}{2}\Delta m\,v_{1}^{% 2}.
  39. Δ m p 1 ρ - Δ m p 2 ρ + Δ m g z 1 - Δ m g z 2 = 1 2 Δ m v 2 2 - 1 2 Δ m v 1 2 \Delta m\,\frac{p_{1}}{\rho}-\Delta m\,\frac{p_{2}}{\rho}+\Delta m\,gz_{1}-% \Delta m\,gz_{2}=\frac{1}{2}\Delta m\,v_{2}^{2}-\frac{1}{2}\Delta m\,v_{1}^{2}
  40. 1 2 Δ m v 1 2 + Δ m g z 1 + Δ m p 1 ρ = 1 2 Δ m v 2 2 + Δ m g z 2 + Δ m p 2 ρ . \frac{1}{2}\Delta m\,v_{1}^{2}+\Delta m\,gz_{1}+\Delta m\,\frac{p_{1}}{\rho}=% \frac{1}{2}\Delta m\,v_{2}^{2}+\Delta m\,gz_{2}+\Delta m\,\frac{p_{2}}{\rho}.
  41. 1 2 v 1 2 + g z 1 + p 1 ρ = 1 2 v 2 2 + g z 2 + p 2 ρ \frac{1}{2}v_{1}^{2}+gz_{1}+\frac{p_{1}}{\rho}=\frac{1}{2}v_{2}^{2}+gz_{2}+% \frac{p_{2}}{\rho}
  42. v 2 2 + g z + p ρ = C \frac{v^{2}}{2}+gz+\frac{p}{\rho}=C
  43. v 2 2 g + z + p ρ g = C \frac{v^{2}}{2g}+z+\frac{p}{\rho g}=C
  44. v = 2 g z , v=\sqrt{{2g}{z}},
  45. h v = v 2 2 g h_{v}=\frac{v^{2}}{2g}
  46. p = p 0 - ρ g z p=p_{0}-\rho gz\,
  47. ψ = p ρ g \psi=\frac{p}{\rho g}
  48. h v + z elevation + ψ = C h_{v}+z\text{elevation}+\psi=C\,
  49. ρ v 2 2 + ρ g z + p = C \frac{\rho v^{2}}{2}+\rho gz+p=C
  50. 0 = Δ M 1 - Δ M 2 = ρ 1 A 1 v 1 Δ t - ρ 2 A 2 v 2 Δ t 0=\Delta M_{1}-\Delta M_{2}=\rho_{1}A_{1}v_{1}\,\Delta t-\rho_{2}A_{2}v_{2}\,\Delta t
  51. 0 = Δ E 1 - Δ E 2 0=\Delta E_{1}-\Delta E_{2}\,
  52. Δ E 1 = [ 1 2 ρ 1 v 1 2 + Ψ 1 ρ 1 + ϵ 1 ρ 1 + p 1 ] A 1 v 1 Δ t \Delta E_{1}=\left[\frac{1}{2}\rho_{1}v_{1}^{2}+\Psi_{1}\rho_{1}+\epsilon_{1}% \rho_{1}+p_{1}\right]A_{1}v_{1}\,\Delta t
  53. Δ E 2 \Delta E_{2}
  54. 0 = Δ E 1 - Δ E 2 0=\Delta E_{1}-\Delta E_{2}
  55. 0 = [ 1 2 ρ 1 v 1 2 + Ψ 1 ρ 1 + ϵ 1 ρ 1 + p 1 ] A 1 v 1 Δ t - [ 1 2 ρ 2 v 2 2 + Ψ 2 ρ 2 + ϵ 2 ρ 2 + p 2 ] A 2 v 2 Δ t 0=\left[\frac{1}{2}\rho_{1}v_{1}^{2}+\Psi_{1}\rho_{1}+\epsilon_{1}\rho_{1}+p_{% 1}\right]A_{1}v_{1}\,\Delta t-\left[\frac{1}{2}\rho_{2}v_{2}^{2}+\Psi_{2}\rho_% {2}+\epsilon_{2}\rho_{2}+p_{2}\right]A_{2}v_{2}\,\Delta t
  56. 0 = [ 1 2 v 1 2 + Ψ 1 + ϵ 1 + p 1 ρ 1 ] ρ 1 A 1 v 1 Δ t - [ 1 2 v 2 2 + Ψ 2 + ϵ 2 + p 2 ρ 2 ] ρ 2 A 2 v 2 Δ t 0=\left[\frac{1}{2}v_{1}^{2}+\Psi_{1}+\epsilon_{1}+\frac{p_{1}}{\rho_{1}}% \right]\rho_{1}A_{1}v_{1}\,\Delta t-\left[\frac{1}{2}v_{2}^{2}+\Psi_{2}+% \epsilon_{2}+\frac{p_{2}}{\rho_{2}}\right]\rho_{2}A_{2}v_{2}\,\Delta t
  57. 1 2 v 2 + Ψ + ϵ + p ρ = constant b \frac{1}{2}v^{2}+\Psi+\epsilon+\frac{p}{\rho}={\rm constant}\equiv b
  58. 1 2 v 2 + Ψ + h = constant b \frac{1}{2}v^{2}+\Psi+h={\rm constant}\equiv b

Bernoulli_number.html

  1. B n B_{n}
  2. 1 1
  3. ± 1 2 \pm\frac{1}{2}
  4. 1 6 \frac{1}{6}
  5. - 1 30 -\frac{1}{30}
  6. 1 42 \frac{1}{42}
  7. - 1 30 -\frac{1}{30}
  8. 5 66 \frac{5}{66}
  9. - 691 2730 -\frac{691}{2730}
  10. 7 6 \frac{7}{6}
  11. - 3617 510 -\frac{3617}{510}
  12. 43867 798 \frac{43867}{798}
  13. - 174611 330 -\frac{174611}{330}
  14. 1 / 2 {1}/{2}
  15. 1 / 6 {1}/{6}
  16. 1 / 30 {1}/{30}
  17. 1 / 42 {1}/{42}
  18. 1 / 30 {1}/{30}
  19. 1 / 2 {1}/{2}
  20. 1 / 2 {1}/{2}
  21. S m ( n ) = k = 1 n k m = 1 m + 2 m + + n m . S_{m}(n)=\sum_{k=1}^{n}k^{m}=1^{m}+2^{m}+\cdots+n^{m}.\,
  22. S m ( n ) = 1 m + 1 k = 0 m ( m + 1 k ) B k n m + 1 - k , S_{m}(n)={1\over{m+1}}\sum_{k=0}^{m}{m+1\choose{k}}B_{k}\;n^{m+1-k},
  23. ( m + 1 k ) {\textstyle\left({{m+1}\atop{k}}\right)}
  24. 1 + 2 + + n = 1 2 ( B 0 n 2 + 2 B 1 n 1 ) = 1 2 ( n 2 + n ) . 1+2+\cdots+n=\frac{1}{2}\left(B_{0}n^{2}+2B_{1}n^{1}\right)=\frac{1}{2}\left(n% ^{2}+n\right).
  25. 1 2 + 2 2 + + n 2 = 1 3 ( B 0 n 3 + 3 B 1 n 2 + 3 B 2 n 1 ) = 1 3 ( n 3 + 3 2 n 2 + 1 2 n ) . 1^{2}+2^{2}+\cdots+n^{2}=\frac{1}{3}\left(B_{0}n^{3}+3B_{1}n^{2}+3B_{2}n^{1}% \right)=\frac{1}{3}\left(n^{3}+\frac{3}{2}n^{2}+\frac{1}{2}n\right).
  26. S m ( n ) = 1 m + 1 k = 0 m ( - 1 ) k ( m + 1 k ) B k n m + 1 - k . S_{m}(n)={1\over{m+1}}\sum_{k=0}^{m}(-1)^{k}{m+1\choose{k}}B_{k}\;n^{m+1-k}.
  27. B n = ( - 1 ) n B n B_{n}=(-1)^{n}B^{\prime}_{n}
  28. B m ( n ) \displaystyle B_{m}(n)
  29. 1 / 2 {1}/{2}
  30. 1 / 2 {1}/{2}
  31. n = 0 : B m \displaystyle n=0:B_{m}
  32. B m ( n ) = k = 0 m v = 0 k ( - 1 ) v ( k v ) ( n + v ) m k + 1 B_{m}(n)=\sum_{k=0}^{m}\sum_{v=0}^{k}(-1)^{v}{\left({{k}\atop{v}}\right)}\frac% {\left(n+v\right)^{m}}{k+1}
  33. n = 0 : B m = k = 0 m v = 0 k ( - 1 ) v ( k v ) v m k + 1 n = 1 : B m = k = 0 m v = 0 k ( - 1 ) v ( k v ) ( v + 1 ) m k + 1 . \begin{aligned}\displaystyle n=0:B_{m}&\displaystyle=\sum_{k=0}^{m}\sum_{v=0}^% {k}(-1)^{v}{\left({{k}\atop{v}}\right)}\frac{v^{m}}{k+1}\\ \displaystyle n=1:B_{m}&\displaystyle=\sum_{k=0}^{m}\sum_{v=0}^{k}(-1)^{v}{% \left({{k}\atop{v}}\right)}\frac{(v+1)^{m}}{k+1}.\end{aligned}
  34. t e n t e t - 1 = m = 0 B m ( n ) t m m ! . \frac{te^{nt}}{e^{t}-1}=\sum_{m=0}^{\infty}B_{m}(n)\frac{t^{m}}{m!}\ .
  35. n = 0 : t e t - 1 = m = 0 B m t m m ! n = 1 : t 1 - e - t = m = 0 B m ( - t ) m m ! . \begin{aligned}\displaystyle n=0:\frac{t}{e^{t}-1}&\displaystyle=\sum_{m=0}^{% \infty}B_{m}\frac{t^{m}}{m!}\\ \displaystyle n=1:\frac{t}{1-e^{-t}}&\displaystyle=\sum_{m=0}^{\infty}B_{m}% \frac{(-t)^{m}}{m!}.\end{aligned}
  36. ( z e z e z - 1 ) x = x n 0 σ n ( x ) z n \left(\frac{ze^{z}}{e^{z}-1}\right)^{x}=x\sum_{n\geq 0}\sigma_{n}(x)z^{n}
  37. k = a b - 1 f ( k ) = a b f ( x ) d x + k = 1 m B k k ! ( f ( k - 1 ) ( b ) - f ( k - 1 ) ( a ) ) + R - ( f , m ) . \sum\limits_{k=a}^{b-1}f(k)=\int_{a}^{b}f(x)\,dx\ +\sum\limits_{k=1}^{m}\frac{% B_{k}}{k!}\left(f^{(k-1)}(b)-f^{(k-1)}(a)\right)+R_{-}(f,m).
  38. k = a + 1 b f ( k ) = a b f ( x ) d x + k = 1 m B k k ! ( f ( k - 1 ) ( b ) - f ( k - 1 ) ( a ) ) + R + ( f , m ) . \sum\limits_{k=a+1}^{b}f(k)=\int_{a}^{b}f(x)\,dx\ +\sum\limits_{k=1}^{m}\frac{% B_{k}}{k!}\left(f^{(k-1)}(b)-f^{(k-1)}(a)\right)+R_{+}(f,m).
  39. a b f ( x ) d x = f ( - 1 ) ( b ) - f ( - 1 ) ( a ) . \int_{a}^{b}f(x)\,dx\ =f^{(-1)}(b)-f^{(-1)}(a).
  40. k = a b f ( k ) = k = 0 m B k k ! ( f ( k - 1 ) ( b ) - f ( k - 1 ) ( a ) ) + R ( f , m ) . \sum\limits_{k=a}^{b}f(k)=\sum\limits_{k=0}^{m}\frac{B_{k}}{k!}\left(f^{(k-1)}% (b)-f^{(k-1)}(a)\right)+R(f,m).
  41. 1 / 2 {1}/{2}
  42. ζ ( s ) \displaystyle\zeta(s)
  43. s k ¯ s^{\overline{k}}
  44. 1 / 2 {1}/{2}
  45. ψ ( z ) ln z - k = 1 B k k z k \psi(z)\sim\ln z-\sum_{k=1}^{\infty}\frac{B_{k}}{kz^{k}}
  46. tan x \displaystyle\tan x
  47. E S n = ( 2 2 n - 2 - 2 4 n - 3 ) Numerator ( B 4 n 4 n ) . ES_{n}=\left(2^{2n-2}-2^{4n-3}\right)\ \,\text{Numerator}\left(\frac{B_{4n}}{4% n}\right).
  48. W n , k = v = 0 k ( - 1 ) v + k ( v + 1 ) n k ! v ! ( k - v ) ! . W_{n,k}=\sum_{v=0}^{k}(-1)^{v+k}\left(v+1\right)^{n}\frac{k!}{v!(k-v)!}\ .
  49. W n , k = k ! { n + 1 k + 1 } . W_{n,k}=k!\left\{{n+1\atop k+1}\right\}.
  50. B n = k = 0 n ( - 1 ) k W n , k k + 1 = k = 0 n 1 k + 1 v = 0 k ( - 1 ) v ( v + 1 ) n ( k v ) . B_{n}=\sum_{k=0}^{n}(-1)^{k}\frac{W_{n,k}}{k+1}\ =\ \sum_{k=0}^{n}\frac{1}{k+1% }\sum_{v=0}^{k}(-1)^{v}\left(v+1\right)^{n}{k\choose v}\ .
  51. B n = n 2 n + 1 - 2 k = 0 n - 1 ( - 2 ) - k W n - 1 , k . B_{n}=\frac{n}{2^{n+1}-2}\sum_{k=0}^{n-1}(-2)^{-k}\;W_{n-1,k}\ .
  52. n + 1 2 n + 2 - 2 × \frac{n+1}{2^{n+2}-2}\times
  53. S ( k , m ) S(k,m)\!
  54. j k = m = 0 k j m ¯ S ( k , m ) j^{k}=\sum_{m=0}^{k}{j^{\underline{m}}}S(k,m)\!
  55. j m ¯ j^{\underline{m}}\!
  56. B k ( j ) B_{k}(j)\!
  57. B k ( j ) = k m = 0 k - 1 ( j m + 1 ) S ( k - 1 , m ) m ! + B k B_{k}(j)=k\sum_{m=0}^{k-1}{j\choose m+1}S(k-1,m)m!+B_{k}\!
  58. B k B_{k}\!
  59. k = 0 , 1 , 2 , k=0,1,2,...\!
  60. ( j m ) = ( j + 1 m + 1 ) - ( j m + 1 ) {j\choose m}={j+1\choose m+1}-{j\choose m+1}\!
  61. j k = B k + 1 ( j + 1 ) - B k + 1 ( j ) k + 1 . j^{k}=\frac{B_{k+1}(j+1)-B_{k+1}(j)}{k+1}.\!
  62. B k ( j ) = n = 0 k ( k n ) B n j k - n . B_{k}(j)=\sum_{n=0}^{k}{{k\choose n}B_{n}j^{k-n}}.\!
  63. ( j m + 1 ) {\textstyle\left({{j}\atop{m+1}}\right)}
  64. ( - 1 ) m m + 1 . \tfrac{(-1)^{m}}{m+1}.
  65. B k = m = 0 k ( - 1 ) m m ! m + 1 S ( k , m ) B_{k}=\sum_{m=0}^{k}(-1)^{m}{\frac{m!}{m+1}}S(k,m)
  66. [ n m ] \textstyle\left[{n\atop m}\right]
  67. 1 m ! k = 0 m ( - 1 ) k [ m + 1 k + 1 ] B k = 1 m + 1 , \frac{1}{m!}\sum_{k=0}^{m}(-1)^{k}\left[{m+1\atop k+1}\right]B_{k}=\frac{1}{m+% 1},
  68. 1 m ! k = 0 m ( - 1 ) k [ m + 1 k + 1 ] B n + k = A n , m . \frac{1}{m!}\sum_{k=0}^{m}(-1)^{k}\left[{m+1\atop k+1}\right]B_{n+k}=A_{n,m}.
  69. n m \textstyle\left\langle{n\atop m}\right\rangle
  70. m = 0 n ( - 1 ) m n m = 2 n + 1 ( 2 n + 1 - 1 ) B n + 1 n + 1 , \sum_{m=0}^{n}(-1)^{m}{\left\langle{n\atop m}\right\rangle}=2^{n+1}(2^{n+1}-1)% \frac{B_{n+1}}{n+1},
  71. m = 0 n ( - 1 ) m n m ( n m ) - 1 = ( n + 1 ) B n . \sum_{m=0}^{n}(-1)^{m}{\left\langle{n\atop m}\right\rangle}{{\left({{n}\atop{m% }}\right)}}^{-1}=(n+1)B_{n}.
  72. N ! = a 1 k = 2 length ( N ) a k ! . N!=a_{1}\prod_{k=2}^{\,\text{length}(N)}a_{k}!.
  73. B n = N node of tree-level n n ! N ! . B_{n}=\sum_{N\ \,\text{node of tree-level}\ n}\frac{n!}{N!}.
  74. B 2 n = ( - 1 ) n + 1 2 ( 2 n ) ! ( 2 π ) 2 n [ 1 + 1 2 2 n + 1 3 2 n + 1 4 2 n + ] . B_{2n}=(-1)^{n+1}\frac{2(2n)!}{(2\pi)^{2n}}\left[1+\frac{1}{2^{2n}}+\frac{1}{3% ^{2n}}+\frac{1}{4^{2n}}+\cdots\;\right].
  75. | B 2 n | 4 π n ( n π e ) 2 n . |B_{2n}|\sim 4\sqrt{\pi n}\left(\frac{n}{\pi e}\right)^{2n}.
  76. b ( s ) = 2 e 1 2 s i π 0 s t s 1 - e 2 π t d t t b(s)=2e^{\frac{1}{2}si\pi}\int_{0}^{\infty}\frac{st^{s}}{1-e^{2\pi t}}\frac{dt% }{t}
  77. p \displaystyle p
  78. π 2 ( 2 2 n - 4 2 n ) B 2 n E 2 n . \pi\ \sim\ 2\left(2^{2n}-4^{2n}\right)\frac{B_{2n}}{E_{2n}}.
  79. B n \displaystyle B_{n}
  80. S n = 2 ( 2 π ) n k = - ( 4 k + 1 ) - n ( k = 0 , - 1 , 1 , - 2 , 2 , ) S_{n}=2\left(\frac{2}{\pi}\right)^{n}\sum_{k=-\infty}^{\infty}\left(4k+1\right% )^{-n}\quad(k=0,-1,1,-2,2,\ldots)
  81. S n = 1 , 1 , 1 2 , 1 3 , 5 24 , 2 15 , 61 720 , 17 315 , 277 8064 , 62 2835 , S_{n}=1,1,\frac{1}{2},\frac{1}{3},\frac{5}{24},\frac{2}{15},\frac{61}{720},% \frac{17}{315},\frac{277}{8064},\frac{62}{2835},\ldots
  82. B n \displaystyle B_{n}
  83. 2 , 4 , 3 , 16 5 , 25 8 , 192 61 , 427 136 , 4352 1385 , 12465 3968 , 158720 50521 , π . 2,4,3,\frac{16}{5},\frac{25}{8},\frac{192}{61},\frac{427}{136},\frac{4352}{138% 5},\frac{12465}{3968},\frac{158720}{50521},\ldots\quad\longrightarrow\pi.
  84. T n = 1 , 1 , 1 , 2 , 5 , 16 , 61 , 272 , 1385 , 7936 , 50521 , 353792 , ( n = 0 , 1 , 2 , 3 , ) T_{n}=1,1,1,2,5,16,61,272,1385,7936,50521,353792,\ldots\quad(n=0,1,2,3,\ldots)
  85. B n \displaystyle B_{n}
  86. tan x \displaystyle\tan x
  87. tan x + sec x = 1 + 1 x + 1 2 x 2 + 1 3 x 3 + 5 24 x 4 + 2 15 x 5 + 61 720 x 6 + \tan x+\sec x=1+1x+\frac{1}{2}x^{2}+\frac{1}{3}x^{3}+\frac{5}{24}x^{4}+\frac{2% }{15}x^{5}+\frac{61}{720}x^{6}+\cdots
  88. B k ( p - 1 ) + b k ( p - 1 ) + b B b b ( mod p ) . \frac{B_{k(p-1)+b}}{k(p-1)+b}\ \equiv\ \frac{B_{b}}{b}\;\;(\mathop{{\rm mod}}p).
  89. m n ( mod p b - 1 ( p - 1 ) ) \scriptstyle m\equiv n\;\;(\mathop{{\rm mod}}p^{b-1}(p-1))
  90. ( 1 - p m - 1 ) B m m ( 1 - p n - 1 ) B n n ( mod p b ) . (1-p^{m-1}){B_{m}\over m}\equiv(1-p^{n-1}){B_{n}\over n}\;\;(\mathop{{\rm mod}% }p^{b}).
  91. ( 1 - p - u ) ζ ( u ) ( 1 - p - v ) ζ ( v ) ( mod p b ) , (1-p^{-u})\zeta(u)\equiv(1-p^{-v})\zeta(v)\;\;(\mathop{{\rm mod}}p^{b}),~{}
  92. a 1 ( mod p - 1 ) \scriptstyle a\not\equiv 1\;\;(\mathop{{\rm mod}}p-1)
  93. p \scriptstyle\mathbb{Z}_{p}\,
  94. ( m + 3 m ) B m = { m + 3 3 - j = 1 m / 6 ( m + 3 m - 6 j ) B m - 6 j , if m 0 ( mod 6 ) ; m + 3 3 - j = 1 ( m - 2 ) / 6 ( m + 3 m - 6 j ) B m - 6 j , if m 2 ( mod 6 ) ; - m + 3 6 - j = 1 ( m - 4 ) / 6 ( m + 3 m - 6 j ) B m - 6 j , if m 4 ( mod 6 ) . {{m+3}\choose{m}}B_{m}=\begin{cases}{{m+3}\over 3}-\sum\limits_{j=1}^{m/6}{m+3% \choose{m-6j}}B_{m-6j},&\mbox{if}~{}\ m\equiv 0\;\;(\mathop{{\rm mod}}6);\\ {{m+3}\over 3}-\sum\limits_{j=1}^{(m-2)/6}{m+3\choose{m-6j}}B_{m-6j},&\mbox{if% }~{}\ m\equiv 2\;\;(\mathop{{\rm mod}}6);\\ -{{m+3}\over 6}-\sum\limits_{j=1}^{(m-4)/6}{m+3\choose{m-6j}}B_{m-6j},&\mbox{% if}~{}\ m\equiv 4\;\;(\mathop{{\rm mod}}6).\end{cases}
  95. B 2 n + ( p - 1 ) | 2 n 1 p B_{2n}+\sum_{(p-1)|2n}\frac{1}{p}
  96. φ k ( n ) = i = 0 n i k - n k 2 \varphi_{k}(n)=\sum_{i=0}^{n}i^{k}-\frac{n^{k}}{2}
  97. 2 B n = m = 0 n ( - 1 ) m 2 m + 1 m ! { n + 1 m + 1 } = 0 ( n > 1 is odd ) 2B_{n}=\sum_{m=0}^{n}\left(-1\right)^{m}\frac{2}{m+1}m!\left\{{n+1\atop m+1}% \right\}=0\quad\left(n>1\ \,\text{is odd}\right)
  98. S n , m = m ! { n m } \textstyle S_{n,m}=m!\left\{{n\atop m}\right\}
  99. m = 1 , 3 , 5 , n 2 m 2 S n , m = m = 2 , 4 , 6 , n 2 m 2 S n , m ( n > 2 is even ) . \sum_{m=1,3,5,\ldots\leq n}\frac{2}{m^{2}}S_{n,m}=\sum_{m=2,4,6,\ldots\leq n}% \frac{2}{m^{2}}S_{n,m}\quad\left(n>2\ \,\text{is even}\right).
  100. R ( x ) = 2 k = 1 k k ¯ x k ( 2 π ) 2 k ( B 2 k / ( 2 k ) ) = 2 k = 1 k k ¯ x k ( 2 π ) 2 k β 2 k . R(x)=2\sum_{k=1}^{\infty}\frac{k^{\overline{k}}x^{k}}{(2\pi)^{2k}\left(B_{2k}/% (2k)\right)}=2\sum_{k=1}^{\infty}\frac{k^{\overline{k}}x^{k}}{(2\pi)^{2k}\beta% _{2k}}.
  101. n k ¯ n^{\overline{k}}
  102. n m = 1 m + 1 ( B 0 n m + 1 - ( m + 1 1 ) B 1 n m + ( m + 1 2 ) B 2 n m - 1 - + ( - 1 ) m ( m + 1 m ) B m n ) \quad\sum n^{m}=\frac{1}{m+1}\left(B_{0}n^{m+1}-{\left({{m+1}\atop{1}}\right)}% B_{1}n^{m}+{\left({{m+1}\atop{2}}\right)}B_{2}n^{m-1}-\cdots+(-1)^{m}{\left({{% m+1}\atop{m}}\right)}B_{m}n\right)
  103. n m \scriptstyle\sum n^{m}
  104. c k ¯ \scriptstyle c^{\underline{k}}
  105. 0 < k n k c = n c + 1 c + 1 + 1 2 n c + k 2 B k k ! c k - 1 ¯ n c - k + 1 . \sum_{0<k\leq n}k^{c}=\frac{n^{c+1}}{c+1}+\frac{1}{2}n^{c}+\sum_{k\geq 2}\frac% {B_{k}}{k!}c^{\underline{k-1}}n^{c-k+1}.
  106. c k - 1 ¯ \scriptstyle c^{\underline{k-1}}
  107. 1 c + 1 \scriptstyle\frac{1}{c+1}
  108. 0 < k n k c = k 0 B k k ! c k - 1 ¯ n c - k + 1 . \sum_{0<k\leq n}k^{c}=\sum_{k\geq 0}\frac{B_{k}}{k!}c^{\underline{k-1}}n^{c-k+% 1}.
  109. a = 1 f χ ( a ) t e a t e f t - 1 = k = 0 B k , χ t k k ! . \sum_{a=1}^{f}\chi(a)\frac{te^{at}}{e^{ft}-1}=\sum_{k=0}^{\infty}B_{k,\chi}% \frac{t^{k}}{k!}.
  110. L ( 1 - k , χ ) = - B k , χ k , L(1-k,\chi)=-\frac{B_{k,\chi}}{k},
  111. 𝐁 k \mathbf{B}^{k}
  112. B k B_{k}
  113. B 1 = - 1 2 \scriptstyle B_{1}=-{1\over 2}
  114. S m ( n ) = 0 n ( 𝐁 + x ) m d x S_{m}(n)=\int_{0}^{n}(\mathbf{B}+x)^{m}\,dx
  115. ( 𝐁 + 1 ) m = B m (\mathbf{B}+1)^{m}=B_{m}
  116. ζ ( n ) = ( - 1 ) n 2 - 1 B n ( 2 π ) n 2 ( n ! ) \zeta(n)=\frac{\left(-1\right)^{\frac{n}{2}-1}B_{n}\left(2\pi\right)^{n}}{2(n!)}
  117. B n = n ! | 1 0 0 1 2 ? 1 0 0 n ? ( n - 1 ) ? 1 0 ( n + 1 ) ? n ? 2 ? 0 | B_{n}=n!\begin{vmatrix}1&0&\cdots&0&1\\ 2?&1&&0&0\\ \vdots&&\ddots&&\vdots\\ n?&(n-1)?&&1&0\\ (n+1)?&n?&\cdots&2?&0\end{vmatrix}
  118. = n ! | 1 0 0 1 1 2 ! 1 0 0 1 n ! 1 ( n - 1 ) ! 1 0 1 ( n + 1 ) ! 1 n ! 1 2 ! 0 | =n!\begin{vmatrix}1&0&\cdots&0&1\\ \frac{1}{2!}&1&&0&0\\ \vdots&&\ddots&&\vdots\\ \frac{1}{n!}&\frac{1}{(n-1)!}&&1&0\\ \frac{1}{(n+1)!}&\frac{1}{n!}&\cdots&\frac{1}{2!}&0\end{vmatrix}
  119. B 2 p = - ( 2 p ) ! 2 2 p - 2 | 1 0 0 0 1 1 3 ! 1 0 0 0 1 5 ! 1 3 ! 1 0 0 1 ( 2 p + 1 ) ! 1 ( 2 p - 1 ) ! 1 ( 2 p - 3 ) ! 1 3 ! 0 | B_{2p}=-\frac{(2p)!}{2^{2p}-2}\begin{vmatrix}1&0&0&\cdots&0&1\\ \frac{1}{3!}&1&0&\cdots&0&0\\ \frac{1}{5!}&\frac{1}{3!}&1&&0&0\\ \vdots&&\ddots&&&\vdots\\ \vdots&&&\ddots&&\vdots\\ \frac{1}{(2p+1)!}&\frac{1}{(2p-1)!}&\frac{1}{(2p-3)!}&\cdots&\frac{1}{3!}&0% \end{vmatrix}
  120. 1 n k = 1 n ( n k ) B k B n - k + B n - 1 = - B n (L. Euler) \frac{1}{n}\sum_{k=1}^{n}{\left({{n}\atop{k}}\right)}B_{k}B_{n-k}+B_{n-1}=-B_{% n}\quad\,\text{(L. Euler)}
  121. k = 0 n ( n + 1 k ) ( n + k + 1 ) B n + k = 0 \sum_{k=0}^{n}{\left({{n+1}\atop{k}}\right)}(n+k+1)B_{n+k}=0
  122. B n = - k = 1 n + 1 ( - 1 ) k k ( n + 1 k ) j = 1 k j n B_{n}=-\sum_{k=1}^{n+1}\frac{(-1)^{k}}{k}{\left({{n+1}\atop{k}}\right)}\sum_{j% =1}^{k}j^{n}
  123. ( - 1 ) m r = 0 m ( m r ) B n + r = ( - 1 ) n s = 0 n ( n s ) B m + s (-1)^{m}\sum_{r=0}^{m}{\left({{m}\atop{r}}\right)}B_{n+r}=(-1)^{n}\sum_{s=0}^{% n}{\left({{n}\atop{s}}\right)}B_{m+s}
  124. H n = 1 k n k - 1 H_{n}=\sum_{1\leq k\leq n}k^{-1}
  125. n 2 k = 2 n - 2 B n - k n - k B k k - k = 2 n - 2 ( n k ) B n - k n - k B k = H n B n (H. Miki, 1978) \frac{n}{2}\sum_{k=2}^{n-2}\frac{B_{n-k}}{n-k}\frac{B_{k}}{k}-\sum_{k=2}^{n-2}% {\left({{n}\atop{k}}\right)}\frac{B_{n-k}}{n-k}B_{k}=H_{n}B_{n}\qquad\,\text{(% H. Miki, 1978)}
  126. ( n + 2 ) k = 2 n - 2 B k B n - k - 2 l = 2 n - 2 ( n + 2 l ) B l B n - l = n ( n + 1 ) B n (n+2)\sum_{k=2}^{n-2}B_{k}B_{n-k}-2\sum_{l=2}^{n-2}{\left({{n+2}\atop{l}}% \right)}B_{l}B_{n-l}=n(n+1)B_{n}
  127. n 2 ( B n - 1 ( x ) + k = 1 n - 1 B k ( x ) k B n - k ( x ) n - k ) - k = 0 n - 1 ( n k ) B n - k n - k B k ( x ) = H n - 1 B n ( x ) . \frac{n}{2}\left(B_{n-1}(x)+\sum_{k=1}^{n-1}\frac{B_{k}(x)}{k}\frac{B_{n-k}(x)% }{n-k}\right)-\sum_{k=0}^{n-1}{\left({{n}\atop{k}}\right)}\frac{B_{n-k}}{n-k}B% _{k}(x)=H_{n-1}B_{n}(x).
  128. k = 0 n ( n k ) B k n - k + 2 = B n + 1 n + 1 \sum_{k=0}^{n}{\left({{n}\atop{k}}\right)}\frac{B_{k}}{n-k+2}=\frac{B_{n+1}}{n% +1}
  129. - 1 + k = 0 n ( n k ) 2 n - k + 1 n - k + 1 B k ( 1 ) = 2 n -1+\sum_{k=0}^{n}{\left({{n}\atop{k}}\right)}\frac{2^{n-k+1}}{n-k+1}B_{k}(1)=2% ^{n}
  130. - 1 + k = 0 n ( n k ) 2 n - k + 1 n - k + 1 B k ( 0 ) = [ n = 0 ] -1+\sum_{k=0}^{n}{\left({{n}\atop{k}}\right)}\frac{2^{n-k+1}}{n-k+1}B_{k}(0)=[% n=0]
  131. ( - 1 ) m + 1 j = 0 k ( k j ) B m + 1 + j m + 1 + j + ( - 1 ) k + 1 j = 0 m B k + 1 + j k + 1 + j = k ! m ! ( k + m + 1 ) ! (-1)^{m+1}\sum_{j=0}^{k}{\left({{k}\atop{j}}\right)}\frac{B_{m+1+j}}{m+1+j}+(-% 1)^{k+1}\sum_{j=0}^{m}\frac{B_{k+1+j}}{k+1+j}=\frac{k!m!}{(k+m+1)!}

Bessel_function.html

  1. x 2 d 2 y d x 2 + x d y d x + ( x 2 - α 2 ) y = 0 x^{2}\frac{d^{2}y}{dx^{2}}+x\frac{dy}{dx}+(x^{2}-\alpha^{2})y=0
  2. J α ( x ) = m = 0 ( - 1 ) m m ! Γ ( m + α + 1 ) ( x 2 ) 2 m + α J_{\alpha}(x)=\sum_{m=0}^{\infty}\frac{(-1)^{m}}{m!\,\Gamma(m+\alpha+1)}{\left% (\frac{x}{2}\right)}^{2m+\alpha}
  3. J - n ( x ) = ( - 1 ) n J n ( x ) . J_{-n}(x)=(-1)^{n}J_{n}(x).\,
  4. J n ( x ) = 1 π 0 π cos ( n τ - x sin ( τ ) ) d τ . J_{n}(x)=\frac{1}{\pi}\int_{0}^{\pi}\cos(n\tau-x\sin(\tau))\,d\tau.
  5. J n ( x ) = 1 2 π - π π e i ( n τ - x sin ( τ ) ) d τ . J_{n}(x)=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{i(n\tau-x\sin(\tau))}\,d\tau.
  6. J α ( x ) = 1 π 0 π cos ( α τ - x sin τ ) d τ - sin ( α π ) π 0 e - x sinh ( t ) - α t d t . J_{\alpha}(x)=\frac{1}{\pi}\int_{0}^{\pi}\cos(\alpha\tau-x\sin\tau)\,d\tau-% \frac{\sin(\alpha\pi)}{\pi}\int_{0}^{\infty}e^{-x\sinh(t)-\alpha t}\,dt.
  7. J α ( x ) = ( x 2 ) α Γ ( α + 1 ) 0 F 1 ( α + 1 ; - x 2 4 ) . J_{\alpha}(x)=\frac{(\frac{x}{2})^{\alpha}}{\Gamma(\alpha+1)}\;_{0}F_{1}(% \alpha+1;-\tfrac{x^{2}}{4}).
  8. J α ( x ) ( x 2 ) α = e - t Γ ( α + 1 ) k = 0 L k ( α ) ( x 2 4 t ) < m t p l > ( k + α k ) t k k ! . \frac{J_{\alpha}(x)}{\left(\frac{x}{2}\right)^{\alpha}}=\frac{e^{-t}}{\Gamma(% \alpha+1)}\sum_{k=0}^{\infty}\frac{L_{k}^{(\alpha)}\left(\frac{x^{2}}{4t}% \right)}{<}mtpl>{{k+\alpha\choose k}}\frac{t^{k}}{k!}.
  9. Y α ( x ) = J α ( x ) cos ( α π ) - J - α ( x ) sin ( α π ) . Y_{\alpha}(x)=\frac{J_{\alpha}(x)\cos(\alpha\pi)-J_{-\alpha}(x)}{\sin(\alpha% \pi)}.
  10. Y n ( x ) = lim α n Y α ( x ) . Y_{n}(x)=\lim_{\alpha\to n}Y_{\alpha}(x).
  11. Y n ( x ) = 1 π 0 π sin ( x sin θ - n θ ) d θ - 1 π 0 [ e n t + ( - 1 ) n e - n t ] e - x sinh t d t . Y_{n}(x)=\frac{1}{\pi}\int_{0}^{\pi}\sin(x\sin\theta-n\theta)\,d\theta-\frac{1% }{\pi}\int_{0}^{\infty}\left[e^{nt}+(-1)^{n}e^{-nt}\right]e^{-x\sinh t}\,dt.
  12. Y - n ( x ) = ( - 1 ) n Y n ( x ) . Y_{-n}(x)=(-1)^{n}Y_{n}(x).\,
  13. H α ( 1 ) ( x ) = J α ( x ) + i Y α ( x ) H_{\alpha}^{(1)}(x)=J_{\alpha}(x)+iY_{\alpha}(x)
  14. H α ( 2 ) ( x ) = J α ( x ) - i Y α ( x ) H_{\alpha}^{(2)}(x)=J_{\alpha}(x)-iY_{\alpha}(x)
  15. H α ( 1 ) ( x ) = J - α ( x ) - e - α π i J α ( x ) i sin ( α π ) H_{\alpha}^{(1)}(x)=\frac{J_{-\alpha}(x)-e^{-\alpha\pi i}J_{\alpha}(x)}{i\sin(% \alpha\pi)}
  16. H α ( 2 ) ( x ) = J - α ( x ) - e α π i J α ( x ) - i sin ( α π ) . H_{\alpha}^{(2)}(x)=\frac{J_{-\alpha}(x)-e^{\alpha\pi i}J_{\alpha}(x)}{-i\sin(% \alpha\pi)}.
  17. H - α ( 1 ) ( x ) = e α π i H α ( 1 ) ( x ) H_{-\alpha}^{(1)}(x)=e^{\alpha\pi i}H_{\alpha}^{(1)}(x)
  18. H - α ( 2 ) ( x ) = e - α π i H α ( 2 ) ( x ) . H_{-\alpha}^{(2)}(x)=e^{-\alpha\pi i}H_{\alpha}^{(2)}(x).
  19. J - ( m + 1 2 ) ( x ) = ( - 1 ) m + 1 Y m + 1 2 ( x ) J_{-(m+\frac{1}{2})}(x)=(-1)^{m+1}Y_{m+\frac{1}{2}}(x)
  20. Y - ( m + 1 2 ) ( x ) = ( - 1 ) m J m + 1 2 ( x ) . Y_{-(m+\frac{1}{2})}(x)=(-1)^{m}J_{m+\frac{1}{2}}(x).
  21. H α ( 1 ) ( x ) = 1 π i - + + i π e x sinh t - α t d t , H_{\alpha}^{(1)}(x)=\frac{1}{\pi i}\int_{-\infty}^{+\infty+i\pi}e^{x\sinh t-% \alpha t}\,dt,
  22. H α ( 2 ) ( x ) = - 1 π i - + - i π e x sinh t - α t d t , H_{\alpha}^{(2)}(x)=-\frac{1}{\pi i}\int_{-\infty}^{+\infty-i\pi}e^{x\sinh t-% \alpha t}\,dt,
  23. I α ( x ) = i - α J α ( i x ) = m = 0 1 m ! Γ ( m + α + 1 ) ( x 2 ) 2 m + α I_{\alpha}(x)=i^{-\alpha}J_{\alpha}(ix)=\sum_{m=0}^{\infty}\frac{1}{m!\,\Gamma% (m+\alpha+1)}\left(\frac{x}{2}\right)^{2m+\alpha}
  24. K α ( x ) = π 2 I - α ( x ) - I α ( x ) sin ( α π ) , K_{\alpha}(x)=\frac{\pi}{2}\frac{I_{-\alpha}(x)-I_{\alpha}(x)}{\sin(\alpha\pi)},
  25. K α ( x ) = π 2 i α + 1 H α ( 1 ) ( i x ) , K_{\alpha}(x)=\frac{\pi}{2}i^{\alpha+1}H_{\alpha}^{(1)}(ix),
  26. x 2 d 2 y d x 2 + x d y d x - ( x 2 + α 2 ) y = 0. x^{2}\frac{d^{2}y}{dx^{2}}+x\frac{dy}{dx}-(x^{2}+\alpha^{2})y=0.
  27. I α ( x ) = 1 π 0 π exp ( x cos ( θ ) ) cos ( α θ ) d θ - sin ( α π ) π 0 exp ( - x cosh t - α t ) d t , I_{\alpha}(x)=\frac{1}{\pi}\int_{0}^{\pi}\exp(x\cos(\theta))\cos(\alpha\theta)% \,d\theta-\frac{\sin(\alpha\pi)}{\pi}\int_{0}^{\infty}\exp(-x\cosh t-\alpha t)% \,dt,
  28. K α ( x ) = 0 exp ( - x cosh t ) cosh ( α t ) d t . K_{\alpha}(x)=\int_{0}^{\infty}\exp(-x\cosh t)\cosh(\alpha t)\,dt.
  29. K 1 3 ( ξ ) = 3 0 exp [ - ξ ( 1 + 4 x 2 3 ) 1 + x 2 3 ] d x K 2 3 ( ξ ) = 1 3 0 3 + 2 x 2 1 + x 2 3 exp [ - ξ ( 1 + 4 x 2 3 ) 1 + x 2 3 ] d x . \begin{aligned}\displaystyle K_{\frac{1}{3}}(\xi)&\displaystyle=\sqrt{3}\,\int% _{0}^{\infty}\,\exp\left[-\xi\left(1+\frac{4x^{2}}{3}\right)\sqrt{1+\frac{x^{2% }}{3}}\,\right]\,dx\\ \displaystyle K_{\frac{2}{3}}(\xi)&\displaystyle=\frac{1}{\sqrt{3}}\,\int_{0}^% {\infty}\,\frac{3+2x^{2}}{\sqrt{1+\frac{x^{2}}{3}}}\exp\left[-\xi\left(1+\frac% {4x^{2}}{3}\right)\sqrt{1+\frac{x^{2}}{3}}\,\right]\,dx.\end{aligned}
  30. x 2 d 2 y d x 2 + 2 x d y d x + [ x 2 - n ( n + 1 ) ] y = 0. x^{2}\frac{d^{2}y}{dx^{2}}+2x\frac{dy}{dx}+[x^{2}-n(n+1)]y=0.
  31. j n ( x ) = π 2 x J n + 1 2 ( x ) , j_{n}(x)=\sqrt{\frac{\pi}{2x}}J_{n+\frac{1}{2}}(x),
  32. y n ( x ) = π 2 x Y n + 1 2 ( x ) = ( - 1 ) n + 1 π 2 x J - n - 1 2 ( x ) . y_{n}(x)=\sqrt{\frac{\pi}{2x}}Y_{n+\frac{1}{2}}(x)=(-1)^{n+1}\sqrt{\frac{\pi}{% 2x}}J_{-n-\frac{1}{2}}(x).
  33. j n ( x ) = ( - x ) n ( 1 x d d x ) n sin ( x ) x , j_{n}(x)=(-x)^{n}\left(\frac{1}{x}\frac{d}{dx}\right)^{n}\,\frac{\sin(x)}{x},
  34. y n ( x ) = - ( - x ) n ( 1 x d d x ) n cos ( x ) x . y_{n}(x)=-(-x)^{n}\left(\frac{1}{x}\frac{d}{dx}\right)^{n}\,\frac{\cos(x)}{x}.
  35. j 0 ( x ) = sin ( x ) x j_{0}(x)=\frac{\sin(x)}{x}
  36. j 1 ( x ) = sin ( x ) x 2 - cos ( x ) x j_{1}(x)=\frac{\sin(x)}{x^{2}}-\frac{\cos(x)}{x}
  37. j 2 ( x ) = ( 3 x 2 - 1 ) sin ( x ) x - 3 cos ( x ) x 2 j_{2}(x)=\left(\frac{3}{x^{2}}-1\right)\frac{\sin(x)}{x}-\frac{3\cos(x)}{x^{2}}
  38. j 3 ( x ) = ( 15 x 3 - 6 x ) sin ( x ) x - ( 15 x 2 - 1 ) cos ( x ) x , j_{3}(x)=\left(\frac{15}{x^{3}}-\frac{6}{x}\right)\frac{\sin(x)}{x}-\left(% \frac{15}{x^{2}}-1\right)\frac{\cos(x)}{x},
  39. y 0 ( x ) = - j - 1 ( x ) = - cos ( x ) x y_{0}(x)=-j_{-1}(x)=-\,\frac{\cos(x)}{x}
  40. y 1 ( x ) = j - 2 ( x ) = - cos ( x ) x 2 - sin ( x ) x y_{1}(x)=j_{-2}(x)=-\,\frac{\cos(x)}{x^{2}}-\frac{\sin(x)}{x}
  41. y 2 ( x ) = - j - 3 ( x ) = ( - 3 x 2 + 1 ) cos ( x ) x - 3 sin ( x ) x 2 y_{2}(x)=-j_{-3}(x)=\left(-\,\frac{3}{x^{2}}+1\right)\frac{\cos(x)}{x}-\frac{3% \sin(x)}{x^{2}}
  42. y 3 ( x ) = j - 4 ( x ) = ( - 15 x 3 + 6 x ) cos ( x ) x - ( 15 x 2 - 1 ) sin ( x ) x . y_{3}\left(x\right)=j_{-4}(x)=\left(-\frac{15}{x^{3}}+\frac{6}{x}\right)\frac{% \cos(x)}{x}-\left(\frac{15}{x^{2}}-1\right)\frac{\sin(x)}{x}.
  43. 1 z cos ( z 2 - 2 z t ) = n = 0 t n n ! j n - 1 ( z ) , \frac{1}{z}\cos\left(\sqrt{z^{2}-2zt}\right)=\sum_{n=0}^{\infty}\frac{t^{n}}{n% !}j_{n-1}(z),
  44. 1 z sin ( z 2 + 2 z t ) = n = 0 ( - t ) n n ! y n - 1 ( z ) . \frac{1}{z}\sin\left(\sqrt{z^{2}+2zt}\right)=\sum_{n=0}^{\infty}\frac{(-t)^{n}% }{n!}y_{n-1}(z).
  45. j n , y n , h n ( 1 ) , h n ( 2 ) j_{n},y_{n},h_{n}^{(1)},h_{n}^{(2)}
  46. n = 0 , ± 1 , ± 2 , n=0,\pm 1,\pm 2,\dots
  47. ( 1 z d d z ) m ( z n + 1 f n ( z ) ) = z n - m + 1 f n - m ( z ) , \left(\frac{1}{z}\frac{d}{dz}\right)^{m}\left(z^{n+1}f_{n}(z)\right)=z^{n-m+1}% f_{n-m}(z),
  48. ( 1 z d d z ) m ( z - n f n ( z ) ) = ( - 1 ) m z - n - m f n + m ( z ) . \left(\frac{1}{z}\frac{d}{dz}\right)^{m}\left(z^{-n}f_{n}(z)\right)=(-1)^{m}z^% {-n-m}f_{n+m}(z).
  49. h n ( 1 ) ( x ) = j n ( x ) + i y n ( x ) h_{n}^{(1)}(x)=j_{n}(x)+iy_{n}(x)\,
  50. h n ( 2 ) ( x ) = j n ( x ) - i y n ( x ) . h_{n}^{(2)}(x)=j_{n}(x)-iy_{n}(x).\,
  51. h n ( 1 ) ( x ) = ( - i ) n + 1 e i x x m = 0 n i m m ! ( 2 x ) m ( n + m ) ! ( n - m ) ! h_{n}^{(1)}(x)=(-i)^{n+1}\frac{e^{ix}}{x}\sum_{m=0}^{n}\frac{i^{m}}{m!(2x)^{m}% }\frac{(n+m)!}{(n-m)!}
  52. h n ( 2 ) h_{n}^{(2)}
  53. j 0 ( x ) = sin ( x ) / x j_{0}(x)=\sin(x)/x
  54. y 0 ( x ) = - cos ( x ) / x y_{0}(x)=-\cos(x)/x
  55. S n ( x ) = x j n ( x ) = π x 2 J n + 1 2 ( x ) S_{n}(x)=xj_{n}(x)=\sqrt{\frac{\pi x}{2}}\,J_{n+\frac{1}{2}}(x)
  56. C n ( x ) = - x y n ( x ) = - π x 2 Y n + 1 2 ( x ) C_{n}(x)=-xy_{n}(x)=-\sqrt{\frac{\pi x}{2}}\,Y_{n+\frac{1}{2}}(x)
  57. ξ n ( x ) = x h n ( 1 ) ( x ) = π x 2 H n + 1 2 ( 1 ) ( x ) = S n ( x ) - i C n ( x ) \xi_{n}(x)=xh_{n}^{(1)}(x)=\sqrt{\frac{\pi x}{2}}\,H_{n+\frac{1}{2}}^{(1)}(x)=% S_{n}(x)-iC_{n}(x)
  58. ζ n ( x ) = x h n ( 2 ) ( x ) = π x 2 H n + 1 2 ( 2 ) ( x ) = S n ( x ) + i C n ( x ) . \zeta_{n}(x)=xh_{n}^{(2)}(x)=\sqrt{\frac{\pi x}{2}}\,H_{n+\frac{1}{2}}^{(2)}(x% )=S_{n}(x)+iC_{n}(x).
  59. x 2 d 2 y d x 2 + [ x 2 - n ( n + 1 ) ] y = 0. x^{2}\frac{d^{2}y}{dx^{2}}+[x^{2}-n(n+1)]y=0.
  60. ψ n , χ n \psi_{n},\chi_{n}
  61. S n , C n S_{n},C_{n}
  62. 0 < z α + 1 0<z\ll\sqrt{\alpha+1}
  63. J α ( z ) 1 Γ ( α + 1 ) ( z 2 ) α J_{\alpha}(z)\sim\frac{1}{\Gamma(\alpha+1)}\left(\frac{z}{2}\right)^{\alpha}
  64. J α ( z ) ( - 1 ) α ( - α ) ! ( 2 z ) α J_{\alpha}(z)\sim\frac{(-1)^{\alpha}}{(-\alpha)!}\left(\frac{2}{z}\right)^{\alpha}
  65. Y α ( z ) { 2 π ( ln ( z 2 ) + γ ) if α = 0 - Γ ( α ) π ( 2 z ) α + 1 Γ ( α + 1 ) ( z 2 ) α cot ( α π ) if α is not a non-positive integer (one term dominates unless α is imaginary) - ( - 1 ) α Γ ( - α ) π ( z 2 ) α if α is a negative integer Y_{\alpha}(z)\sim\begin{cases}\frac{2}{\pi}\left(\ln\left(\frac{z}{2}\right)+% \gamma\right)&\,\text{if }\alpha=0\\ \\ -\frac{\Gamma(\alpha)}{\pi}\left(\frac{2}{z}\right)^{\alpha}+\frac{1}{\Gamma(% \alpha+1)}\left(\frac{z}{2}\right)^{\alpha}\cot(\alpha\pi)&\,\text{if }\alpha% \,\text{ is not a non-positive integer (one term dominates unless }\alpha\,% \text{ is imaginary)}\\ \\ -\frac{(-1)^{\alpha}\Gamma(-\alpha)}{\pi}\left(\frac{z}{2}\right)^{\alpha}&\,% \text{if }\alpha\,\text{ is a negative integer}\end{cases}
  66. x | α 2 - 1 4 | x\gg\left|\alpha^{2}-\tfrac{1}{4}\right|
  67. J α ( z ) = 2 π z ( cos ( z - α π 2 - π 4 ) + e | Im ( z ) | O ( | z | - 1 ) ) for | arg z | < π Y α ( z ) = 2 π z ( sin ( z - α π 2 - π 4 ) + e | Im ( z ) | O ( | z | - 1 ) ) for | arg z | < π . \begin{aligned}\displaystyle J_{\alpha}(z)&\displaystyle=\sqrt{\frac{2}{\pi z}% }\left(\cos\left(z-\frac{\alpha\pi}{2}-\frac{\pi}{4}\right)+e^{|\operatorname{% Im}(z)|}O(|z|^{-1})\right)&&\displaystyle\,\text{ for }|\arg z|<\pi\\ \displaystyle Y_{\alpha}(z)&\displaystyle=\sqrt{\frac{2}{\pi z}}\left(\sin% \left(z-\frac{\alpha\pi}{2}-\frac{\pi}{4}\right)+e^{|\operatorname{Im}(z)|}O(|% z|^{-1})\right)&&\displaystyle\,\text{ for }|\arg z|<\pi.\end{aligned}
  68. J 0 ( z ) - 2 π z cos ( z + π 4 ) J_{0}(z)\approx\sqrt{\frac{-2}{\pi z}}\cos\left(z+\frac{\pi}{4}\right)
  69. J 0 ( z ) 2 π z cos ( z - π 4 ) . J_{0}(z)\approx\sqrt{\frac{2}{\pi z}}\cos\left(z-\frac{\pi}{4}\right).
  70. H α ( 1 ) ( z ) \displaystyle H_{\alpha}^{(1)}(z)
  71. H α ( 1 ) ( z e i m π ) H_{\alpha}^{(1)}(ze^{im\pi})
  72. H α ( 2 ) ( z e i m π ) H_{\alpha}^{(2)}(ze^{im\pi})
  73. J α ( z ) \displaystyle J_{\alpha}(z)
  74. I α ( z ) e z 2 π z ( 1 - 4 α 2 - 1 8 z + ( 4 α 2 - 1 ) ( 4 α 2 - 9 ) 2 ! ( 8 z ) 2 - ( 4 α 2 - 1 ) ( 4 α 2 - 9 ) ( 4 α 2 - 25 ) 3 ! ( 8 z ) 3 + ) for | arg z | < π 2 , I_{\alpha}(z)\sim\frac{e^{z}}{\sqrt{2\pi z}}\left(1-\frac{4\alpha^{2}-1}{8z}+% \frac{(4\alpha^{2}-1)(4\alpha^{2}-9)}{2!(8z)^{2}}-\frac{(4\alpha^{2}-1)(4% \alpha^{2}-9)(4\alpha^{2}-25)}{3!(8z)^{3}}+\cdots\right)\,\text{ for }|\arg z|% <\tfrac{\pi}{2},
  75. K α ( z ) π 2 z e - z ( 1 + 4 α 2 - 1 8 z + ( 4 α 2 - 1 ) ( 4 α 2 - 9 ) 2 ! ( 8 z ) 2 + ( 4 α 2 - 1 ) ( 4 α 2 - 9 ) ( 4 α 2 - 25 ) 3 ! ( 8 z ) 3 + ) for | arg z | < 3 π 2 . K_{\alpha}(z)\sim\sqrt{\frac{\pi}{2z}}e^{-z}\left(1+\frac{4\alpha^{2}-1}{8z}+% \frac{(4\alpha^{2}-1)(4\alpha^{2}-9)}{2!(8z)^{2}}+\frac{(4\alpha^{2}-1)(4% \alpha^{2}-9)(4\alpha^{2}-25)}{3!(8z)^{3}}+\cdots\right)\,\text{ for }|\arg z|% <\tfrac{3\pi}{2}.
  76. I 1 2 ( z ) = 2 π z sinh ( z ) e z 2 π z for | arg z | < π 2 , K 1 2 ( z ) = π 2 z e - z \begin{aligned}\displaystyle I_{\frac{1}{2}}(z)&\displaystyle=\sqrt{\frac{2}{% \pi z}}\sinh(z)\sim\frac{e^{z}}{\sqrt{2\pi z}}&&\displaystyle\,\text{ for }|% \arg z|<\tfrac{\pi}{2},\\ \displaystyle K_{\frac{1}{2}}(z)&\displaystyle=\sqrt{\frac{\pi}{2z}}e^{-z}\end% {aligned}
  77. 0 < | z | α + 1 0<|z|\ll\sqrt{\alpha+1}
  78. I α ( z ) 1 Γ ( α + 1 ) ( z 2 ) α I_{\alpha}(z)\sim\frac{1}{\Gamma(\alpha+1)}\left(\frac{z}{2}\right)^{\alpha}
  79. K α ( z ) { - ln ( z 2 ) - γ if α = 0 Γ ( α ) 2 ( 2 z ) α if α > 0. K_{\alpha}(z)\sim\begin{cases}-\ln\left(\frac{z}{2}\right)-\gamma&\,\text{if }% \alpha=0\\ \\ \frac{\Gamma(\alpha)}{2}\left(\frac{2}{z}\right)^{\alpha}&\,\text{if }\alpha>0% .\end{cases}
  80. e ( x 2 ) ( t - 1 / t ) = n = - J n ( x ) t n , e^{(\frac{x}{2})(t-1/t)}=\sum_{n=-\infty}^{\infty}J_{n}(x)t^{n},\!
  81. e i z cos ( ϕ ) = n = - i n J n ( z ) e i n ϕ , e^{iz\cos(\phi)}=\sum_{n=-\infty}^{\infty}i^{n}J_{n}(z)e^{in\phi},\!
  82. e ± i z sin ( ϕ ) = J 0 ( z ) + 2 n = 1 J 2 n ( z ) cos ( 2 n ϕ ) ± 2 i n = 0 J 2 n + 1 ( z ) sin ( [ 2 n + 1 ] ϕ ) , e^{\pm iz\sin(\phi)}=J_{0}(z)+2\sum_{n=1}^{\infty}J_{2n}(z)\cos(2n\phi)\pm 2i% \sum_{n=0}^{\infty}J_{2n+1}(z)\sin([2n+1]\phi),\!
  83. f ( z ) = a 0 ν J ν ( z ) + 2 k = 1 a k ν J ν + k ( z ) f(z)=a_{0}^{\nu}J_{\nu}(z)+2\cdot\sum_{k=1}a_{k}^{\nu}J_{\nu+k}(z)\!
  84. a k 0 = 1 2 π i | z | = c f ( z ) O k ( z ) d z , a_{k}^{0}=\frac{1}{2\pi i}\int_{|z|=c}f(z)O_{k}(z)\,dz,\!
  85. f ( z ) = k = 0 a k ν J ν + 2 k ( z ) f(z)=\sum_{k=0}a_{k}^{\nu}J_{\nu+2k}(z)\!
  86. a k ν = 2 ( ν + 2 k ) 0 f ( z ) J ν + 2 k ( z ) z d z a_{k}^{\nu}=2(\nu+2k)\int_{0}^{\infty}f(z)\frac{J_{\nu+2k}(z)}{z}\,dz\!
  87. 0 J α ( z ) J β ( z ) d z z = 2 π sin ( π 2 ( α - β ) ) α 2 - β 2 . \int_{0}^{\infty}J_{\alpha}(z)J_{\beta}(z)\frac{dz}{z}=\frac{2}{\pi}\frac{\sin% \left(\frac{\pi}{2}(\alpha-\beta)\right)}{\alpha^{2}-\beta^{2}}.
  88. f ( z ) = k = 0 a k J ν + k ( z ) , f(z)=\sum_{k=0}a_{k}J_{\nu+k}(z),
  89. { k = 0 a k J ν + k } ( s ) = 1 1 + s 2 k = 0 a k ( s + 1 + s 2 ) ν + k \mathcal{L}\left\{\sum_{k=0}a_{k}J_{\nu+k}\right\}(s)=\frac{1}{\sqrt{1+s^{2}}}% \sum_{k=0}\frac{a_{k}}{(s+\sqrt{1+s^{2}})^{\nu+k}}
  90. k = 0 a k ξ ν + k = 1 + ξ 2 2 ξ { f } ( 1 - ξ 2 2 ξ ) , \sum_{k=0}a_{k}\xi^{\nu+k}=\frac{1+\xi^{2}}{2\xi}\mathcal{L}\{f\}\left(\frac{1% -\xi^{2}}{2\xi}\right),
  91. { f } \mathcal{L}\{f\}
  92. J ν ( z ) \displaystyle J_{\nu}(z)
  93. 2 α x Z α ( x ) = Z α - 1 ( x ) + Z α + 1 ( x ) \frac{2\alpha}{x}Z_{\alpha}(x)=Z_{\alpha-1}(x)+Z_{\alpha+1}(x)\!
  94. 2 d Z α d x = Z α - 1 ( x ) - Z α + 1 ( x ) 2\frac{dZ_{\alpha}}{dx}=Z_{\alpha-1}(x)-Z_{\alpha+1}(x)\!
  95. ( 1 x d d x ) m [ x α Z α ( x ) ] = x α - m Z α - m ( x ) , \left(\frac{1}{x}\frac{d}{dx}\right)^{m}\left[x^{\alpha}Z_{\alpha}(x)\right]=x% ^{\alpha-m}Z_{\alpha-m}(x),
  96. ( 1 x d d x ) m [ Z α ( x ) x α ] = ( - 1 ) m Z α + m ( x ) x α + m . \left(\frac{1}{x}\frac{d}{dx}\right)^{m}\left[\frac{Z_{\alpha}(x)}{x^{\alpha}}% \right]=(-1)^{m}\frac{Z_{\alpha+m}(x)}{x^{\alpha+m}}.
  97. e ( x 2 ) ( t + 1 / t ) = n = - I n ( x ) t n , e^{(\frac{x}{2})(t+1/t)}=\sum_{n=-\infty}^{\infty}I_{n}(x)t^{n},\!
  98. e z cos ( θ ) = I 0 ( z ) + 2 n = 1 I n ( z ) cos ( n θ ) . e^{z\cos(\theta)}=I_{0}(z)+2\sum_{n=1}^{\infty}I_{n}(z)\cos(n\theta).\!
  99. C α - 1 ( x ) - C α + 1 ( x ) = 2 α x C α ( x ) C_{\alpha-1}(x)-C_{\alpha+1}(x)=\frac{2\alpha}{x}C_{\alpha}(x)\!
  100. C α - 1 ( x ) + C α + 1 ( x ) = 2 d C α d x C_{\alpha-1}(x)+C_{\alpha+1}(x)=2\frac{dC_{\alpha}}{dx}\!
  101. 0 1 x J α ( x u α , m ) J α ( x u α , n ) d x = δ m , n 2 [ J α + 1 ( u α , m ) ] 2 = δ m , n 2 [ J α ( u α , m ) ] 2 , \int_{0}^{1}xJ_{\alpha}(xu_{\alpha,m})J_{\alpha}(xu_{\alpha,n})\,dx=\frac{% \delta_{m,n}}{2}[J_{\alpha+1}(u_{\alpha,m})]^{2}=\frac{\delta_{m,n}}{2}[J_{% \alpha}^{\prime}(u_{\alpha,m})]^{2},\!
  102. 0 1 x 2 j α ( x u α , m ) j α ( x u α , n ) d x = δ m , n 2 [ j α + 1 ( u α , m ) ] 2 . \int_{0}^{1}x^{2}j_{\alpha}(xu_{\alpha,m})j_{\alpha}(xu_{\alpha,n})\,dx=\frac{% \delta_{m,n}}{2}[j_{\alpha+1}(u_{\alpha,m})]^{2}.\!
  103. f ϵ ( x ) = ϵ rect ( x - 1 ϵ ) f_{\epsilon}(x)=\epsilon\ \mathrm{rect}\left(\frac{x-1}{\epsilon}\right)
  104. 0 k J α ( k x ) g ϵ ( k ) d k = f ϵ ( x ) \int_{0}^{\infty}kJ_{\alpha}(kx)g_{\epsilon}(k)dk=f_{\epsilon}(x)
  105. 0 k J α ( k x ) J α ( k ) d k = δ ( x - 1 ) \int_{0}^{\infty}kJ_{\alpha}(kx)J_{\alpha}(k)dk=\delta(x-1)
  106. 0 x J α ( u x ) J α ( v x ) d x = 1 u δ ( u - v ) \int_{0}^{\infty}xJ_{\alpha}(ux)J_{\alpha}(vx)\,dx=\frac{1}{u}\delta(u-v)\!
  107. 0 x 2 j α ( u x ) j α ( v x ) d x = π 2 u 2 δ ( u - v ) \int_{0}^{\infty}x^{2}j_{\alpha}(ux)j_{\alpha}(vx)\,dx=\frac{\pi}{2u^{2}}% \delta(u-v)\!
  108. A α ( x ) d B α d x - d A α d x B α ( x ) = C α x , A_{\alpha}(x)\frac{dB_{\alpha}}{dx}-\frac{dA_{\alpha}}{dx}B_{\alpha}(x)=\frac{% C_{\alpha}}{x},\!
  109. J α ( x ) d Y α d x - d J α d x Y α ( x ) = 2 π x , J_{\alpha}(x)\frac{dY_{\alpha}}{dx}-\frac{dJ_{\alpha}}{dx}Y_{\alpha}(x)=\frac{% 2}{\pi x},\!
  110. I α ( x ) d K α d x - d I α d x K α ( x ) = - 1 x . I_{\alpha}(x)\frac{dK_{\alpha}}{dx}-\frac{dI_{\alpha}}{dx}K_{\alpha}(x)=-\frac% {1}{x}.\!
  111. λ - ν J ν ( λ z ) = n = 0 1 n ! ( ( 1 - λ 2 ) z 2 ) n J ν + n ( z ) \lambda^{-\nu}J_{\nu}(\lambda z)=\sum_{n=0}^{\infty}\frac{1}{n!}\left(\frac{(1% -\lambda^{2})z}{2}\right)^{n}J_{\nu+n}(z)
  112. J J
  113. Y Y
  114. λ - ν I ν ( λ z ) = n = 0 1 n ! ( ( λ 2 - 1 ) z 2 ) n I ν + n ( z ) \lambda^{-\nu}I_{\nu}(\lambda z)=\sum_{n=0}^{\infty}\frac{1}{n!}\left(\frac{(% \lambda^{2}-1)z}{2}\right)^{n}I_{\nu+n}(z)
  115. λ - ν K ν ( λ z ) = n = 0 ( - 1 ) n n ! ( ( λ 2 - 1 ) z 2 ) n K ν + n ( z ) . \lambda^{-\nu}K_{\nu}(\lambda z)=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\left(% \frac{(\lambda^{2}-1)z}{2}\right)^{n}K_{\nu+n}(z).
  116. K 1 2 ( z ) = π 2 e - z z - 1 2 , z > 0 I - 1 2 ( z ) = 2 π z cosh ( z ) I 1 2 ( z ) = 2 π z sinh ( z ) I ν ( z ) = k = 0 z k k ! J ν + k ( z ) J ν ( z ) = k = 0 ( - 1 ) k z k k ! I ν + k ( z ) I ν ( λ z ) = λ ν k = 0 ( ( λ 2 - 1 ) z 2 ) k k ! I ν + k ( z ) I ν ( z 1 + z 2 ) = k = - I ν - k ( z 1 ) I k ( z 2 ) J ν ( z 1 ± z 2 ) = k = - J ν k ( z 1 ) J k ( z 2 ) I ν ( z ) = z 2 ν ( I ν - 1 ( z ) - I ν + 1 ( z ) ) J ν ( z ) = z 2 ν ( J ν - 1 ( z ) + J ν + 1 ( z ) ) J ν ( z ) = { 1 2 ( J ν - 1 ( z ) - J ν + 1 ( z ) ) ν 0 - J 1 ( z ) ν = 0 I ν ( z ) = { 1 2 ( I ν - 1 ( z ) + I ν + 1 ( z ) ) ν 0 I 1 ( z ) ν = 0 ( z 2 ) ν = Γ ( ν ) k = 0 I ν + 2 k ( z ) ( ν + 2 k ) ( - ν k ) = Γ ( ν ) k = 0 ( - 1 ) k J ν + 2 k ( z ) ( ν + 2 k ) ( - ν k ) = Γ ( ν + 1 ) k = 0 ( z 2 ) k k ! J ν + k ( z ) 1 = n = 0 ( 2 n + 1 ) j n ( z ) 2 sin ( 2 z ) 2 z = n = 0 ( - 1 ) n ( 2 n + 1 ) j n ( z ) 2 \begin{aligned}\displaystyle K_{\frac{1}{2}}(z)&\displaystyle=\sqrt{\frac{\pi}% {2}}e^{-z}z^{-\tfrac{1}{2}},\qquad z>0\\ \displaystyle I_{-\frac{1}{2}}(z)&\displaystyle=\sqrt{\frac{2}{\pi z}}\cosh(z)% \\ \displaystyle I_{\frac{1}{2}}(z)&\displaystyle=\sqrt{\frac{2}{\pi z}}\sinh(z)% \\ \displaystyle I_{\nu}(z)&\displaystyle=\sum_{k=0}\frac{z^{k}}{k!}J_{\nu+k}(z)% \\ \displaystyle J_{\nu}(z)&\displaystyle=\sum_{k=0}(-1)^{k}\frac{z^{k}}{k!}I_{% \nu+k}(z)\\ \displaystyle I_{\nu}(\lambda z)&\displaystyle=\lambda^{\nu}\sum_{k=0}\frac{% \left((\lambda^{2}-1)\frac{z}{2}\right)^{k}}{k!}I_{\nu+k}(z)\\ \displaystyle I_{\nu}(z_{1}+z_{2})&\displaystyle=\sum_{k=-\infty}^{\infty}I_{% \nu-k}(z_{1})I_{k}(z_{2})\\ \displaystyle J_{\nu}(z_{1}\pm z_{2})&\displaystyle=\sum_{k=-\infty}^{\infty}J% _{\nu\mp k}(z_{1})J_{k}(z_{2})\\ \displaystyle I_{\nu}(z)&\displaystyle=\tfrac{z}{2\nu}\left(I_{\nu-1}(z)-I_{% \nu+1}(z)\right)\\ \displaystyle J_{\nu}(z)&\displaystyle=\tfrac{z}{2\nu}\left(J_{\nu-1}(z)+J_{% \nu+1}(z)\right)\\ \displaystyle J_{\nu}^{\prime}(z)&\displaystyle=\begin{cases}\tfrac{1}{2}\left% (J_{\nu-1}(z)-J_{\nu+1}(z)\right)&\nu\neq 0\\ -J_{1}(z)&\nu=0\end{cases}\\ \displaystyle I_{\nu}^{\prime}(z)&\displaystyle=\begin{cases}\tfrac{1}{2}\left% (I_{\nu-1}(z)+I_{\nu+1}(z)\right)&\nu\neq 0\\ I_{1}(z)&\nu=0\end{cases}\\ \displaystyle\left(\tfrac{z}{2}\right)^{\nu}&\displaystyle=\Gamma(\nu)\sum_{k=% 0}I_{\nu+2k}(z)(\nu+2k){-\nu\choose k}=\Gamma(\nu)\sum_{k=0}(-1)^{k}J_{\nu+2k}% (z)(\nu+2k){-\nu\choose k}=\Gamma(\nu+1)\sum_{k=0}\frac{\left(\tfrac{z}{2}% \right)^{k}}{k!}J_{\nu+k}(z)\\ \displaystyle 1&\displaystyle=\sum_{n=0}^{\infty}(2n+1)j_{n}(z)^{2}\\ \displaystyle\frac{\sin(2z)}{2z}&\displaystyle=\sum_{n=0}^{\infty}(-1)^{n}(2n+% 1)j_{n}(z)^{2}\end{aligned}

Beta_decay.html

  1. A A
  2. Z Z
  3. A A
  4. Z Z
  5. 1 / 3 {1}/{3}
  6. 2 / 3 {2}/{3}
  7. Q Q
  8. Q Q
  9. Q Q
  10. Q Q
  11. Q Q
  12. Q = [ m N ( X Z A ) - m N ( X Z + 1 A ) - m e - m ν ¯ e ] c 2 Q=\left[m_{N}\left({}^{A}_{Z}\mathrm{X}\right)-m_{N}\left({}^{A}_{Z+1}\mathrm{% X^{\prime}}\right)-m_{e}-m_{\overline{\nu}_{e}}\right]c^{2}
  13. m N ( X Z A ) m_{N}\left({}^{A}_{Z}\mathrm{X}\right)
  14. m e m_{e}
  15. m ν ¯ e m_{\overline{\nu}_{e}}
  16. m m
  17. m ( X Z A ) c 2 = m N ( X Z A ) c 2 + Z m e c 2 - i = 1 Z B i m\left({}^{A}_{Z}\mathrm{X}\right)c^{2}=m_{N}\left({}^{A}_{Z}\mathrm{X}\right)% c^{2}+Zm_{e}c^{2}-\sum_{i=1}^{Z}B_{i}
  18. Z Z
  19. Q = [ m ( X Z A ) - m ( X Z + 1 A ) ] c 2 Q=\left[m\left({}^{A}_{Z}\mathrm{X}\right)-m\left({}^{A}_{Z+1}\mathrm{X^{% \prime}}\right)\right]c^{2}
  20. Q Q
  21. Q = [ m N ( X Z A ) - m N ( X Z - 1 A ) - m e - m ν e ] c 2 Q=\left[m_{N}\left({}^{A}_{Z}\mathrm{X}\right)-m_{N}\left({}^{A}_{Z-1}\mathrm{% X^{\prime}}\right)-m_{e}-m_{\nu_{e}}\right]c^{2}
  22. Q = [ m ( X Z A ) - m ( X Z - 1 A ) - 2 m e ] c 2 Q=\left[m\left({}^{A}_{Z}\mathrm{X}\right)-m\left({}^{A}_{Z-1}\mathrm{X^{% \prime}}\right)-2m_{e}\right]c^{2}
  23. Q Q
  24. Q = [ m N ( X Z A ) + m e - m N ( X Z - 1 A ) - m ν e ] c 2 Q=\left[m_{N}\left({}^{A}_{Z}\mathrm{X}\right)+m_{e}-m_{N}\left({}^{A}_{Z-1}% \mathrm{X^{\prime}}\right)-m_{\nu_{e}}\right]c^{2}
  25. Q = [ m ( X Z A ) - m ( X Z - 1 A ) ] c 2 - B n Q=\left[m\left({}^{A}_{Z}\mathrm{X}\right)-m\left({}^{A}_{Z-1}\mathrm{X^{% \prime}}\right)\right]c^{2}-B_{n}
  26. A A
  27. Z Z
  28. A A
  29. A A
  30. ( A , Z ) (A,Z)
  31. ( A , Z 1 ) (A,Z−1)
  32. ( A , Z + 1 ) (A,Z+1)
  33. ( A , Z ) (A,Z)
  34. A A
  35. A A
  36. A A
  37. A A
  38. L L
  39. L > 0 L>0
  40. J J
  41. L L
  42. Δ J = L - 1 , L , L + 1 ; Δ π = ( - 1 ) L , \Delta J=L-1,L,L+1;\Delta\pi=(-1)^{L},
  43. Δ π = 1 Δπ=1
  44. 1 −1
  45. J J
  46. L L
  47. J J
  48. S = 0 S=0
  49. Δ J = 0 \Delta J=0
  50. Δ L = 0 \Delta L=0
  51. 𝒪 F = G V a τ ^ a ± \mathcal{O}_{F}=G_{V}\sum_{a}\hat{\tau}_{a\pm}
  52. G V G_{V}
  53. τ ± \tau_{\pm}
  54. a a
  55. S = 1 S=1
  56. Δ J = 0 , ± 1 \Delta J=0,\pm 1
  57. 𝒪 G T = G A a σ ^ a τ ^ a ± \mathcal{O}_{GT}=G_{A}\sum_{a}\hat{\sigma}_{a}\hat{\tau}_{a\pm}
  58. G A G_{A}
  59. σ \sigma
  60. N ( T ) N(T)
  61. N ( T ) = C L ( T ) F ( Z , T ) p E ( Q - T ) 2 N(T)=C_{L}(T)F(Z,T)pE(Q-T)^{2}
  62. T T
  63. F ( Z , T ) F(Z,T)
  64. p = p=
  65. Q Q
  66. Q Q
  67. F ( Z , T ) = 2 ( 1 + S ) Γ ( 1 + 2 S ) 2 ( 2 p ρ ) 2 S - 2 e π η | Γ ( S + i η ) | 2 , F(Z,T)=\frac{2(1+S)}{\Gamma(1+2S)^{2}}(2p\rho)^{2S-2}e^{\pi\eta}|\Gamma(S+i% \eta)|^{2},
  68. S = S=
  69. η = ± α Z E / p c η=±αZE/pc
  70. F ( Z , T ) 2 π η 1 - e - 2 π η . F(Z,T)\approx\frac{2\pi\eta}{1-e^{-2\pi\eta}}.
  71. Q Q
  72. m / e m/e
  73. m / e m/e

Beta_sheet.html

  1. C \mathrm{C^{\prime}}
  2. C α i \mathrm{C^{\alpha}}_{i}
  3. C i + 2 α \mathrm{C^{\alpha}_{i+2}}
  4. C i α \mathrm{C^{\alpha}_{i}}
  5. C j α \mathrm{C^{\alpha}_{j}}
  6. C i α \mathrm{C^{\alpha}_{i}}
  7. C j α \mathrm{C^{\alpha}_{j}}
  8. i i
  9. j - 1 j-1
  10. j + 1 j+1
  11. j j

Betelgeuse.html

  1. M \begin{smallmatrix}M_{\odot}\end{smallmatrix}
  2. M \begin{smallmatrix}M_{\odot}\end{smallmatrix}
  3. M \begin{smallmatrix}M_{\odot}\end{smallmatrix}

Bézier_curve.html

  1. 𝐁 ( t ) = 𝐏 0 + t ( 𝐏 1 - 𝐏 0 ) = ( 1 - t ) 𝐏 0 + t 𝐏 1 , 0 t 1 \mathbf{B}(t)=\mathbf{P}_{0}+t(\mathbf{P}_{1}-\mathbf{P}_{0})=(1-t)\mathbf{P}_% {0}+t\mathbf{P}_{1}\mbox{ , }~{}0\leq t\leq 1
  2. 𝐁 ( t ) = ( 1 - t ) [ ( 1 - t ) 𝐏 0 + t 𝐏 1 ] + t [ ( 1 - t ) 𝐏 1 + t 𝐏 2 ] , 0 t 1 \mathbf{B}(t)=(1-t)[(1-t)\mathbf{P}_{0}+t\mathbf{P}_{1}]+t[(1-t)\mathbf{P}_{1}% +t\mathbf{P}_{2}]\mbox{ , }~{}0\leq t\leq 1
  3. 𝐁 ( t ) = ( 1 - t ) 2 𝐏 0 + 2 ( 1 - t ) t 𝐏 1 + t 2 𝐏 2 , 0 t 1. \mathbf{B}(t)=(1-t)^{2}\mathbf{P}_{0}+2(1-t)t\mathbf{P}_{1}+t^{2}\mathbf{P}_{2% }\mbox{ , }~{}0\leq t\leq 1.
  4. 𝐁 ( t ) = 2 ( 1 - t ) ( 𝐏 1 - 𝐏 0 ) + 2 t ( 𝐏 2 - 𝐏 1 ) . \mathbf{B}^{\prime}(t)=2(1-t)(\mathbf{P}_{1}-\mathbf{P}_{0})+2t(\mathbf{P}_{2}% -\mathbf{P}_{1})\,.
  5. 𝐁 ′′ ( t ) = 2 ( 𝐏 2 - 2 𝐏 1 + 𝐏 0 ) . \mathbf{B}^{\prime\prime}(t)=2(\mathbf{P}_{2}-2\mathbf{P}_{1}+\mathbf{P}_{0})\,.
  6. 𝐁 ( t ) = ( 1 - t ) 𝐁 𝐏 0 , 𝐏 1 , 𝐏 2 ( t ) + t 𝐁 𝐏 1 , 𝐏 2 , 𝐏 3 ( t ) , 0 t 1. \mathbf{B}(t)=(1-t)\mathbf{B}_{\mathbf{P}_{0},\mathbf{P}_{1},\mathbf{P}_{2}}(t% )+t\mathbf{B}_{\mathbf{P}_{1},\mathbf{P}_{2},\mathbf{P}_{3}}(t)\mbox{ , }~{}0% \leq t\leq 1.
  7. 𝐁 ( t ) = ( 1 - t ) 3 𝐏 0 + 3 ( 1 - t ) 2 t 𝐏 1 + 3 ( 1 - t ) t 2 𝐏 2 + t 3 𝐏 3 , 0 t 1. \mathbf{B}(t)=(1-t)^{3}\mathbf{P}_{0}+3(1-t)^{2}t\mathbf{P}_{1}+3(1-t)t^{2}% \mathbf{P}_{2}+t^{3}\mathbf{P}_{3}\mbox{ , }~{}0\leq t\leq 1.
  8. 𝐁 ( t ) = 3 ( 1 - t ) 2 ( 𝐏 1 - 𝐏 0 ) + 6 ( 1 - t ) t ( 𝐏 2 - 𝐏 1 ) + 3 t 2 ( 𝐏 3 - 𝐏 2 ) . \mathbf{B}^{\prime}(t)=3(1-t)^{2}(\mathbf{P}_{1}-\mathbf{P}_{0})+6(1-t)t(% \mathbf{P}_{2}-\mathbf{P}_{1})+3t^{2}(\mathbf{P}_{3}-\mathbf{P}_{2})\,.
  9. 𝐁 ′′ ( t ) = 6 ( 1 - t ) ( 𝐏 2 - 2 𝐏 1 + 𝐏 0 ) + 6 t ( 𝐏 3 - 2 𝐏 2 + 𝐏 1 ) . \mathbf{B}^{\prime\prime}(t)=6(1-t)(\mathbf{P}_{2}-2\mathbf{P}_{1}+\mathbf{P}_% {0})+6t(\mathbf{P}_{3}-2\mathbf{P}_{2}+\mathbf{P}_{1})\,.
  10. 𝐁 𝐏 0 𝐏 1 𝐏 n \mathbf{B}_{\mathbf{P}_{0}\mathbf{P}_{1}\ldots\mathbf{P}_{n}}
  11. 𝐁 𝐏 0 ( t ) = 𝐏 0 , and \mathbf{B}_{\mathbf{P}_{0}}(t)=\mathbf{P}_{0}\,\text{, and}
  12. 𝐁 ( t ) = 𝐁 𝐏 0 𝐏 1 𝐏 n ( t ) = ( 1 - t ) 𝐁 𝐏 0 𝐏 1 𝐏 n - 1 ( t ) + t 𝐁 𝐏 1 𝐏 2 𝐏 n ( t ) \mathbf{B}(t)=\mathbf{B}_{\mathbf{P}_{0}\mathbf{P}_{1}\ldots\mathbf{P}_{n}}(t)% =(1-t)\mathbf{B}_{\mathbf{P}_{0}\mathbf{P}_{1}\ldots\mathbf{P}_{n-1}}(t)+t% \mathbf{B}_{\mathbf{P}_{1}\mathbf{P}_{2}\ldots\mathbf{P}_{n}}(t)
  13. 𝐁 ( t ) = \displaystyle\mathbf{B}(t)=
  14. ( n i ) \scriptstyle{n\choose i}
  15. 𝐁 𝐏 0 𝐏 1 𝐏 2 𝐏 3 𝐏 4 𝐏 5 ( t ) = 𝐁 ( t ) = \displaystyle\mathbf{B}_{\mathbf{P}_{0}\mathbf{P}_{1}\mathbf{P}_{2}\mathbf{P}_% {3}\mathbf{P}_{4}\mathbf{P}_{5}}(t)=\mathbf{B}(t)=
  16. 𝐁 ( t ) = i = 0 n b i , n ( t ) 𝐏 i , 0 t 1 \mathbf{B}(t)=\sum_{i=0}^{n}b_{i,n}(t)\mathbf{P}_{i},\quad 0\leq t\leq 1
  17. b i , n ( t ) = ( n i ) t i ( 1 - t ) n - i , i = 0 , , n b_{i,n}(t)={n\choose i}t^{i}(1-t)^{n-i},\quad i=0,\ldots,n
  18. ( n i ) \scriptstyle{n\choose i}
  19. 𝐂 i n {}^{n}{\mathbf{C}}_{i}
  20. 𝐂 i n {\mathbf{C}_{i}}^{n}
  21. ( n i ) = n ! i ! ( n - i ) ! {n\choose i}=\frac{n!}{i!(n-i)!}
  22. 𝐁 ( t ) = j = 0 n t j 𝐂 j \mathbf{B}(t)=\sum_{j=0}^{n}{t^{j}\mathbf{C}_{j}}
  23. 𝐂 j = n ! ( n - j ) ! i = 0 j ( - 1 ) i + j 𝐏 i i ! ( j - i ) ! = m = 0 j - 1 ( n - m ) i = 0 j ( - 1 ) i + j 𝐏 i i ! ( j - i ) ! . \mathbf{C}_{j}=\frac{n!}{(n-j)!}\sum_{i=0}^{j}\frac{(-1)^{i+j}\mathbf{P}_{i}}{% i!(j-i)!}=\prod_{m=0}^{j-1}(n-m)\sum_{i=0}^{j}\frac{(-1)^{i+j}\mathbf{P}_{i}}{% i!(j-i)!}.
  24. 𝐂 j \mathbf{C}_{j}
  25. 𝐁 ( t ) \mathbf{B}(t)
  26. 4 ( 2 - 1 ) 3 \textstyle\frac{4\left(\sqrt{2}-1\right)}{3}
  27. 4 3 tan ( t / 4 ) \textstyle\frac{4}{3}\tan(t/4)
  28. 𝐏 k = k n + 1 𝐏 k - 1 + ( 1 - k n + 1 ) 𝐏 k \mathbf{P}^{\prime}_{k}=\tfrac{k}{n+1}\mathbf{P}_{k-1}+\left(1-\tfrac{k}{n+1}% \right)\mathbf{P}_{k}
  29. 𝐁 ( t ) = n i = 0 n - 1 b i , n - 1 ( t ) ( 𝐏 i + 1 - 𝐏 i ) \mathbf{B}^{\prime}(t)=n\sum_{i=0}^{n-1}b_{i,n-1}(t)(\mathbf{P}_{i+1}-\mathbf{% P}_{i})
  30. 𝐁 ( t ) = ( 1 - t ) 𝐁 ( t ) + t 𝐁 ( t ) \mathbf{B}(t)=(1-t)\mathbf{B}(t)+t\mathbf{B}(t)
  31. 𝐛 i , n ( t ) 𝐏 i \mathbf{b}_{i,n}(t)\mathbf{P}_{i}
  32. ( 1 - t ) 2 𝐏 0 + 2 ( 1 - t ) t 𝐏 1 + t 2 𝐏 2 \displaystyle(1-t)^{2}\mathbf{P}_{0}+2(1-t)t\mathbf{P}_{1}+t^{2}\mathbf{P}_{2}
  33. ( n + 1 i ) ( 1 - t ) 𝐛 i , n = ( n i ) 𝐛 i , n + 1 ( 1 - t ) 𝐛 i , n = n + 1 - i n + 1 𝐛 i , n + 1 ( n + 1 i + 1 ) t 𝐛 i , n = ( n i ) 𝐛 i + 1 , n + 1 t 𝐛 i , n = i + 1 n + 1 𝐛 i + 1 , n + 1 𝐁 ( t ) = ( 1 - t ) i = 0 n 𝐛 i , n ( t ) 𝐏 i + t i = 0 n 𝐛 i , n ( t ) 𝐏 i = i = 0 n n + 1 - i n + 1 𝐛 i , n + 1 ( t ) 𝐏 i + i = 0 n i + 1 n + 1 𝐛 i + 1 , n + 1 ( t ) 𝐏 i = i = 0 n + 1 ( i n + 1 𝐏 i - 1 + n + 1 - i n + 1 𝐏 i ) 𝐛 i , n + 1 ( t ) = i = 0 n + 1 𝐛 i , n + 1 ( t ) 𝐏 i \begin{aligned}\displaystyle{n+1\choose i}(1-t)\mathbf{b}_{i,n}&\displaystyle=% {n\choose i}\mathbf{b}_{i,n+1}\\ \displaystyle\Rightarrow(1-t)\mathbf{b}_{i,n}&\displaystyle=\frac{n+1-i}{n+1}% \mathbf{b}_{i,n+1}\\ \displaystyle{n+1\choose i+1}t\mathbf{b}_{i,n}&\displaystyle={n\choose i}% \mathbf{b}_{i+1,n+1}\\ \displaystyle\Rightarrow t\mathbf{b}_{i,n}&\displaystyle=\frac{i+1}{n+1}% \mathbf{b}_{i+1,n+1}\\ \displaystyle\mathbf{B}(t)&\displaystyle=(1-t)\sum_{i=0}^{n}\mathbf{b}_{i,n}(t% )\mathbf{P}_{i}+t\sum_{i=0}^{n}\mathbf{b}_{i,n}(t)\mathbf{P}_{i}\\ &\displaystyle=\sum_{i=0}^{n}\frac{n+1-i}{n+1}\mathbf{b}_{i,n+1}(t)\mathbf{P}_% {i}+\sum_{i=0}^{n}\frac{i+1}{n+1}\mathbf{b}_{i+1,n+1}(t)\mathbf{P}_{i}\\ &\displaystyle=\sum_{i=0}^{n+1}\left(\frac{i}{n+1}\mathbf{P}_{i-1}+\frac{n+1-i% }{n+1}\mathbf{P}_{i}\right)\mathbf{b}_{i,n+1}(t)\\ &\displaystyle=\sum_{i=0}^{n+1}\mathbf{b}_{i,n+1}(t)\mathbf{P^{\prime}}_{i}% \end{aligned}
  34. 𝐏 - 1 \mathbf{P}_{-1}
  35. 𝐏 n + 1 \mathbf{P}_{n+1}
  36. 𝐏 i = i n + 1 𝐏 i - 1 + n + 1 - i n + 1 𝐏 i , i = 0 , , n + 1 \mathbf{P^{\prime}}_{i}=\frac{i}{n+1}\mathbf{P}_{i-1}+\frac{n+1-i}{n+1}\mathbf% {P}_{i},\quad i=0,\ldots,n+1
  37. 𝐏 i , r = j = 0 n 𝐏 j ( n j ) ( r i - j ) ( n + r i ) \mathbf{P}_{i,r}=\sum_{j=0}^{n}\mathbf{P}_{j}{\textstyle\left({{n}\atop{j}}% \right)}\frac{{\textstyle\left({{r}\atop{i-j}}\right)}}{{\textstyle\left({{n+r% }\atop{i}}\right)}}
  38. lim 𝐫 𝐑 𝐫 = 𝐁 \mathbf{\lim_{r\to\infty}R_{r}}=\mathbf{B}
  39. 𝐁 ( t ) = i = 0 n b i , n ( t ) 𝐏 i w i i = 0 n b i , n ( t ) w i \mathbf{B}(t)=\frac{\sum_{i=0}^{n}b_{i,n}(t)\mathbf{P}_{i}w_{i}}{\sum_{i=0}^{n% }b_{i,n}(t)w_{i}}
  40. 𝐁 ( t ) = i = 0 n ( n i ) t i ( 1 - t ) n - i 𝐏 i w i i = 0 n ( n i ) t i ( 1 - t ) n - i w i . \mathbf{B}(t)=\frac{\sum_{i=0}^{n}{n\choose i}t^{i}(1-t)^{n-i}\mathbf{P}_{i}w_% {i}}{\sum_{i=0}^{n}{n\choose i}t^{i}(1-t)^{n-i}w_{i}}.

Bézout's_identity.html

  1. a x + b y = d ax+by=d
  2. | x | < | b d | |x|<\left|\frac{b}{d}\right|
  3. | y | < | a d | . |y|<\left|\frac{a}{d}\right|.
  4. ( x + k b gcd ( a , b ) , y - k a gcd ( a , b ) ) , \left(x+k\frac{b}{\gcd(a,b)},\ y-k\frac{a}{\gcd(a,b)}\right),
  5. k k
  6. | x | < | b gcd ( a , b ) | and | y | < | a gcd ( a , b ) | . |x|<\left|\frac{b}{\gcd(a,b)}\right|\quad\,\text{and}\quad|y|<\left|\frac{a}{% \gcd(a,b)}\right|.
  7. ( q , r ) (q,r)
  8. c = d q + r c=dq+r
  9. 12 × \color b l u e - 10 + 42 × \color b l u e 3 = 6 12 × \color r e d - 3 + 42 × \color r e d 1 = 6 12 × \color r e d 4 + 42 × \color r e d - 1 = 6 12 × \color b l u e 11 + 42 × \color b l u e - 3 = 6 12 × \color b l u e 18 + 42 × \color b l u e - 5 = 6 \begin{aligned}\displaystyle\vdots\\ \displaystyle 12&\displaystyle\times\color{blue}{-10}&\displaystyle+\;\;42&% \displaystyle\times\color{blue}{3}&\displaystyle=6\\ \displaystyle 12&\displaystyle\times\color{red}{-3}&\displaystyle+\;\;42&% \displaystyle\times\color{red}{1}&\displaystyle=6\\ \displaystyle 12&\displaystyle\times\color{red}{4}&\displaystyle+\;\;42&% \displaystyle\times\color{red}{-1}&\displaystyle=6\\ \displaystyle 12&\displaystyle\times\color{blue}{11}&\displaystyle+\;\;42&% \displaystyle\times\color{blue}{-3}&\displaystyle=6\\ \displaystyle 12&\displaystyle\times\color{blue}{18}&\displaystyle+\;\;42&% \displaystyle\times\color{blue}{-5}&\displaystyle=6\\ \displaystyle\vdots\end{aligned}
  10. gcd ( a 1 , a 2 , , a n ) = d \gcd(a_{1},a_{2},\ldots,a_{n})=d
  11. x 1 , x 2 , , x n x_{1},x_{2},\ldots,x_{n}
  12. d = a 1 x 1 + a 2 x 2 + + a n x n , d=a_{1}x_{1}+a_{2}x_{2}+\cdots+a_{n}x_{n},

Biconditional_elimination.html

  1. ( P Q ) (P\leftrightarrow Q)
  2. ( P Q ) (P\to Q)
  3. ( Q P ) (Q\to P)
  4. ( P Q ) ( P Q ) \frac{(P\leftrightarrow Q)}{\therefore(P\to Q)}
  5. ( P Q ) ( Q P ) \frac{(P\leftrightarrow Q)}{\therefore(Q\to P)}
  6. ( P Q ) (P\leftrightarrow Q)
  7. ( P Q ) (P\to Q)
  8. ( Q P ) (Q\to P)
  9. ( P Q ) ( P Q ) (P\leftrightarrow Q)\vdash(P\to Q)
  10. ( P Q ) ( Q P ) (P\leftrightarrow Q)\vdash(Q\to P)
  11. \vdash
  12. ( P Q ) (P\to Q)
  13. ( Q P ) (Q\to P)
  14. ( P Q ) (P\leftrightarrow Q)
  15. ( P Q ) ( P Q ) (P\leftrightarrow Q)\to(P\to Q)
  16. ( P Q ) ( Q P ) (P\leftrightarrow Q)\to(Q\to P)
  17. P P
  18. Q Q

Biconditional_introduction.html

  1. P Q P\to Q
  2. Q P Q\to P
  3. P Q P\leftrightarrow Q
  4. P Q , Q P P Q \frac{P\to Q,Q\to P}{\therefore P\leftrightarrow Q}
  5. P Q P\to Q
  6. Q P Q\to P
  7. P Q P\leftrightarrow Q
  8. ( P Q ) , ( Q P ) ( P Q ) (P\to Q),(Q\to P)\vdash(P\leftrightarrow Q)
  9. \vdash
  10. P Q P\leftrightarrow Q
  11. P Q P\to Q
  12. Q P Q\to P
  13. ( ( P Q ) and ( Q P ) ) ( P Q ) ((P\to Q)\and(Q\to P))\to(P\leftrightarrow Q)
  14. P P
  15. Q Q

Big_Bang_nucleosynthesis.html

  1. t T 2 = ( 0.74 s MeV 2 ) × ( 10.75 / g * ) 1 / 2 tT^{2}=(0.74\mathrm{\ s\ MeV^{2}})\times(10.75/g_{*})^{1/2}

Big_O_notation.html

  1. f ( x ) = O ( g ( x ) ) as x f(x)=O(g(x))\,\text{ as }x\to\infty\,
  2. | f ( x ) | M | g ( x ) | for all x x 0 . |f(x)|\leq\;M|g(x)|\,\text{ for all }x\geq x_{0}.
  3. f ( x ) = O ( g ( x ) ) as x a f(x)=O(g(x))\,\text{ as }x\to a\,
  4. | f ( x ) | M | g ( x ) | for | x - a | < δ . |f(x)|\leq\;M|g(x)|\,\text{ for }|x-a|<\delta.
  5. f ( x ) = O ( g ( x ) ) as x a f(x)=O(g(x))\,\text{ as }x\to a\,
  6. lim sup x a | f ( x ) g ( x ) | < . \limsup_{x\to a}\left|\frac{f(x)}{g(x)}\right|<\infty.
  7. f ( x ) = 6 x 4 - 2 x 3 + 5 f(x)=6x^{4}-2x^{3}+5
  8. | f ( x ) | M | g ( x ) | |f(x)|\leq\;M|g(x)|
  9. | 6 x 4 - 2 x 3 + 5 | \displaystyle|6x^{4}-2x^{3}+5|
  10. | 6 x 4 - 2 x 3 + 5 | 13 x 4 . |6x^{4}-2x^{3}+5|\leq 13\,x^{4}.
  11. T ( n ) = O ( n 2 ) \ T(n)=O(n^{2})\,
  12. T ( n ) O ( n 2 ) T(n)\in O(n^{2})\,
  13. O O
  14. O O
  15. x x
  16. e x \displaystyle e^{x}
  17. x x
  18. f ( n ) = 9 log n + 5 ( log n ) 3 + 3 n 2 + 2 n 3 = O ( n 3 ) , as n . f(n)=9\log n+5(\log n)^{3}+3n^{2}+2n^{3}=O(n^{3})\,,\qquad\,\text{as }n\to% \infty\,\!.
  19. c n c^{n}
  20. log ( n c ) = c log n \log(n^{c})=c\log n
  21. 2 n 2^{n}
  22. 3 n 3^{n}
  23. c 2 n 2 c^{2}n^{2}
  24. c 2 c^{2}
  25. c 2 n 2 O ( n 2 ) c^{2}n^{2}\in O(n^{2})
  26. 2 n 2^{n}
  27. 2 c n = ( 2 c ) n 2^{cn}=(2^{c})^{n}
  28. 2 n 2^{n}
  29. f 1 O ( g 1 ) and f 2 O ( g 2 ) f 1 f 2 O ( g 1 g 2 ) f_{1}\in O(g_{1})\,\text{ and }f_{2}\in O(g_{2})\,\Rightarrow f_{1}f_{2}\in O(% g_{1}g_{2})\,
  30. f O ( g ) O ( f g ) f\cdot O(g)\subset O(fg)
  31. f 1 O ( g 1 ) and f 2 O ( g 2 ) f 1 + f 2 O ( | g 1 | + | g 2 | ) f_{1}\in O(g_{1})\,\text{ and }f_{2}\in O(g_{2})\,\Rightarrow f_{1}+f_{2}\in O% (|g_{1}|+|g_{2}|)\,
  32. f 1 O ( g ) and f 2 O ( g ) f 1 + f 2 O ( g ) f_{1}\in O(g)\,\text{ and }f_{2}\in O(g)\Rightarrow f_{1}+f_{2}\in O(g)
  33. O ( g ) O(g)
  34. f + O ( g ) O ( f + g ) f+O(g)\in O(f+g)
  35. O ( k g ) = O ( g ) \ O(kg)=O(g)
  36. f O ( g ) k f O ( g ) . f\in O(g)\Rightarrow kf\in O(g).
  37. f ( x ) f(\vec{x})
  38. g ( x ) g(\vec{x})
  39. n \mathbb{R}^{n}
  40. f ( x ) is O ( g ( x ) ) as x f(\vec{x})\,\text{ is }O(g(\vec{x}))\,\text{ as }\vec{x}\to\infty
  41. M C > 0 such that for all x with x i M for some i , | f ( x ) | C | g ( x ) | . \exists M\,\exists C>0\,\text{ such that for all }\vec{x}\,\text{ with }x_{i}% \geq M\,\text{ for some }i,|f(\vec{x})|\leq C|g(\vec{x})|.
  42. x i M x_{i}\geq M
  43. i i
  44. x M \|\vec{x}\|\geq M
  45. x \|\vec{x}\|
  46. f ( n , m ) = n 2 + m 3 + O ( n + m ) as n , m f(n,m)=n^{2}+m^{3}+O(n+m)\,\text{ as }n,m\to\infty\,
  47. ( n , m ) M : | g ( n , m ) | C ( n + m ) , \forall\|(n,m)\|\geq M:|g(n,m)|\leq C(n+m),
  48. f ( n , m ) = n 2 + m 3 + g ( n , m ) . f(n,m)=n^{2}+m^{3}+g(n,m).\,
  49. x \vec{x}
  50. f ( n , m ) = O ( n m ) as n , m f(n,m)=O(n^{m})\,\text{ as }n,m\to\infty\,
  51. C M n m \exists C\,\exists M\,\forall n\,\forall m\dots
  52. m : f ( n , m ) = O ( n m ) as n \forall m\colon f(n,m)=O(n^{m})\,\text{ as }n\to\infty
  53. m C M n \forall m\,\exists C\,\exists M\,\forall n\dots
  54. g ( x ) = h ( x ) + O ( f ( x ) ) g(x)=h(x)+O(f(x))\,
  55. g ( x ) - h ( x ) O ( f ( x ) ) . g(x)-h(x)\in O(f(x))\,.
  56. 55 n 3 + 2 n + 10 55n^{3}+2n+10
  57. T ( n ) = 55 n 3 + O ( n 2 ) . T(n)=55n^{3}+O(n^{2}).
  58. f ( m ) = O ( m n ) , f(m)=O(m^{n})\,,
  59. g ( n ) = O ( m n ) . g(n)\,\,=O(m^{n})\,.
  60. g ( x ) O ( f ( x ) ) . g(x)\in O(f(x))\,.
  61. g O ( f ) , g\in O(f)\,,
  62. n n\to\infty
  63. ( n + 1 ) 2 = n 2 + O ( n ) (n+1)^{2}=n^{2}+O(n)
  64. ( n + O ( n 1 / 2 ) ) ( n + O ( log n ) ) 2 = n 3 + O ( n 5 / 2 ) (n+O(n^{1/2}))(n+O(\log n))^{2}=n^{3}+O(n^{5/2})
  65. n O ( 1 ) = O ( e n ) . n^{O(1)}=O(e^{n}).
  66. f ( n ) = O ( 1 ) f(n)=O(1)\,
  67. g ( n ) = O ( e n ) g(n)=O(e^{n})\,
  68. n f ( n ) = g ( n ) n^{f(n)}=g(n)
  69. n O ( 1 ) = O ( e n ) n^{O(1)}=O(e^{n})\,
  70. O ( e n ) = n O ( 1 ) O(e^{n})=n^{O(1)}\,
  71. O ( 1 ) O(1)\,
  72. ( - 1 ) n (-1)^{n}
  73. O ( log log n ) O(\log\log n)\,
  74. O ( log n ) O(\log n)\,
  75. O ( log c n ) , c > 1 O(\log^{c}n),\;c>1\,
  76. O ( n c ) , 0 < c < 1 O(n^{c}),\;0<c<1\,
  77. O ( n ) O(n)\,
  78. O ( n log * n ) O(n\log^{*}n)\,
  79. log * ( n ) = { 0 , if n 1 1 + log * ( log n ) , if n > 1 \log^{*}(n)=\begin{cases}0,&\,\text{if }n\leq 1\\ 1+\log^{*}(\log n),&\,\text{if }n>1\end{cases}
  80. O ( n log n ) = O ( log n ! ) O(n\log n)=O(\log n!)\,
  81. O ( n 2 ) O(n^{2})\,
  82. O ( n c ) , c > 1 O(n^{c}),\;c>1
  83. L n [ α , c ] , 0 < α < 1 = L_{n}[\alpha,c],\;0<\alpha<1=\,
  84. e ( c + o ( 1 ) ) ( ln n ) α ( ln ln n ) 1 - α e^{(c+o(1))(\ln n)^{\alpha}(\ln\ln n)^{1-\alpha}}
  85. O ( c n ) , c > 1 O(c^{n}),\;c>1
  86. O ( n ! ) O(n!)\,
  87. O ( n n ! ) O(n\cdot n!)\,
  88. f ( n ) = O ( n ! ) f(n)=O(n!)\,
  89. f ( n ) = O ( n n ) f(n)=O\left(n^{n}\right)
  90. k > 0 k>0
  91. c > 0 c>0
  92. O ( n c ( log n ) k ) O(n^{c}(\log n)^{k})
  93. O ( n c + ε ) O(n^{c+\varepsilon})
  94. ε > 0 \varepsilon>0
  95. f ( x ) f(x)
  96. g ( x ) g(x)
  97. f ( x ) = o ( g ( x ) ) f(x)=o(g(x))
  98. f ( x ) o ( g ( x ) ) f(x)\in o(g(x))
  99. g ( x ) g(x)
  100. f ( x ) f(x)
  101. f ( x ) f(x)
  102. g ( x ) g(x)
  103. n n→∞
  104. ϵ \epsilon
  105. | f ( n ) | ϵ | g ( n ) | for all n N . |f(n)|\leq\epsilon|g(n)|\qquad\,\text{for all }n\geq N~{}.
  106. ϵ \epsilon
  107. lim x f ( x ) g ( x ) = 0. \lim_{x\to\infty}\frac{f(x)}{g(x)}=0.
  108. 2 x o ( x 2 ) 2x\in o(x^{2})\,\!
  109. 2 x 2 o ( x 2 ) 2x^{2}\not\in o(x^{2})
  110. 1 / x o ( 1 ) 1/x\in o(1)
  111. c o ( f ) = o ( f ) c\cdot o(f)=o(f)
  112. c 0 c\not=0
  113. o ( f ) o ( g ) o ( f g ) o(f)o(g)\subseteq o(fg)
  114. o ( o ( f ) ) o ( f ) o(o(f))\subseteq o(f)
  115. o ( f ) O ( f ) o(f)\subset O(f)
  116. f ( x ) f(x)
  117. o ( g ( x ) ) o(g(x))
  118. f ( x ) = o ( g ( x ) ) f(x)=o(g(x))
  119. f ( x ) = Ω ( g ( x ) ) ( x a ) , f(x)=\Omega(g(x))\ (x\rightarrow a),
  120. a a
  121. \infty
  122. - -\infty
  123. f f
  124. g g
  125. a a
  126. g g
  127. Ω \Omega
  128. f ( x ) = Ω ( g ( x ) ) ( x ) lim sup x | f ( x ) g ( x ) | > 0 f(x)=\Omega(g(x))\ (x\rightarrow\infty)\;\Leftrightarrow\;\limsup_{x\to\infty}% \left|\frac{f(x)}{g(x)}\right|>0
  129. f ( x ) = Ω ( g ( x ) ) f(x)=\Omega(g(x))
  130. f ( x ) = o ( g ( x ) ) f(x)=o(g(x))
  131. Ω R \Omega_{R}
  132. Ω L \Omega_{L}
  133. f ( x ) = Ω R ( g ( x ) ) ( x ) lim sup x f ( x ) g ( x ) > 0 f(x)=\Omega_{R}(g(x))\ (x\rightarrow\infty)\;\Leftrightarrow\;\limsup_{x\to% \infty}\frac{f(x)}{g(x)}>0
  134. f ( x ) = Ω L ( g ( x ) ) ( x ) lim inf x f ( x ) g ( x ) < 0 f(x)=\Omega_{L}(g(x))\ (x\rightarrow\infty)\;\Leftrightarrow\;\liminf_{x\to% \infty}\frac{f(x)}{g(x)}<0
  135. f ( x ) = Ω R ( g ( x ) ) f(x)=\Omega_{R}(g(x))
  136. f ( x ) < o ( g ( x ) ) f(x)<o(g(x))
  137. f ( x ) = Ω L ( g ( x ) ) f(x)=\Omega_{L}(g(x))
  138. f ( x ) > o ( g ( x ) ) f(x)>o(g(x))
  139. Ω R \Omega_{R}
  140. Ω + \Omega_{+}
  141. Ω L \Omega_{L}
  142. Ω - \Omega_{-}
  143. Ω , Ω + , Ω - \Omega,\Omega_{+},\Omega_{-}
  144. f ( x ) = Ω ± ( g ( x ) ) f(x)=\Omega_{\pm}(g(x))
  145. f ( x ) = Ω + ( g ( x ) ) f(x)=\Omega_{+}(g(x))
  146. f ( x ) = Ω - ( g ( x ) ) f(x)=\Omega_{-}(g(x))
  147. sin x = Ω ( 1 ) ( x ) , \sin x=\Omega(1)\ (x\rightarrow\infty),
  148. sin x = Ω ± ( 1 ) ( x ) . \sin x=\Omega_{\pm}(1)\ (x\rightarrow\infty).
  149. sin x + 1 = Ω ( 1 ) ( x ) , \sin x+1=\Omega(1)\ (x\rightarrow\infty),
  150. sin x + 1 = Ω + ( 1 ) ( x ) ; \sin x+1=\Omega_{+}(1)\ (x\rightarrow\infty);
  151. sin x + 1 Ω - ( 1 ) ( x ) . \sin x+1\not=\Omega_{-}(1)\ (x\rightarrow\infty).
  152. Ω \Omega
  153. f ( x ) = Ω ( g ( x ) ) g ( x ) = O ( f ( x ) ) f(x)=\Omega(g(x))\Leftrightarrow g(x)=O(f(x))
  154. Ω \Omega
  155. n n
  156. f ( n ) O ( g ( n ) ) f(n)\in O(g(n))
  157. f ( n ) = O ( g ( n ) ) f(n)=O(g(n))
  158. f f
  159. g g
  160. | f ( n ) | k | g ( n ) | |f(n)|\leq k\cdot|g(n)|
  161. k > 0 n 0 n > n 0 | f ( n ) | k | g ( n ) | \exists k>0\;\exists n_{0}\;\forall n>n_{0}\;|f(n)|\leq k\cdot|g(n)|
  162. k > 0 n 0 n > n 0 f ( n ) k g ( n ) \exists k>0\;\exists n_{0}\;\forall n>n_{0}\;f(n)\leq k\cdot g(n)
  163. f ( n ) Ω ( g ( n ) ) f(n)\in\Omega(g(n))
  164. f ( n ) = Ω ( g ( n ) ) f(n)=\Omega(g(n))
  165. f f
  166. g g
  167. f f
  168. g g
  169. f ( n ) k g ( n ) f(n)\geq k\cdot g(n)
  170. f ( n ) k g ( n ) f(n)\geq k\cdot g(n)
  171. k > 0 n 0 n > n 0 f ( n ) k g ( n ) \exists k>0\;\forall n_{0}\;\exists n>n_{0}\;f(n)\geq k\cdot g(n)
  172. k > 0 n 0 n > n 0 f ( n ) k g ( n ) \exists k>0\;\exists n_{0}\;\forall n>n_{0}\;f(n)\geq k\cdot g(n)
  173. f ( n ) Θ ( g ( n ) ) f(n)\in\Theta(g(n))
  174. f f
  175. g g
  176. k 1 g ( n ) f ( n ) k 2 g ( n ) k_{1}\cdot g(n)\leq f(n)\leq k_{2}\cdot g(n)
  177. k 1 > 0 k 2 > 0 n 0 n > n 0 \exists k_{1}>0\;\exists k_{2}>0\;\exists n_{0}\;\forall n>n_{0}
  178. k 1 g ( n ) f ( n ) k 2 g ( n ) k_{1}\cdot g(n)\leq f(n)\leq k_{2}\cdot g(n)
  179. f ( n ) o ( g ( n ) ) f(n)\in o(g(n))
  180. f ( n ) = o ( g ( n ) ) f(n)=o(g(n))
  181. f f
  182. g g
  183. | f ( n ) | k | g ( n ) | |f(n)|\leq k\cdot|g(n)|
  184. k k
  185. k > 0 n 0 n > n 0 | f ( n ) | k | g ( n ) | \forall k>0\;\exists n_{0}\;\forall n>n_{0}\;|f(n)|\leq k\cdot|g(n)|
  186. f ( n ) ω ( g ( n ) ) f(n)\in\omega(g(n))
  187. f f
  188. g g
  189. | f ( n ) | k | g ( n ) | |f(n)|\geq k\cdot|g(n)|
  190. k k
  191. k > 0 n 0 n > n 0 | f ( n ) | k | g ( n ) | \forall k>0\;\exists n_{0}\;\forall n>n_{0}\ |f(n)|\geq k\cdot|g(n)|
  192. f ( n ) g ( n ) f(n)\sim g(n)\!
  193. f f
  194. g g
  195. f ( n ) / g ( n ) 1 f(n)/g(n)\to 1
  196. ε > 0 n 0 n > n 0 | f ( n ) g ( n ) - 1 | < ε \forall\varepsilon>0\;\exists n_{0}\;\forall n>n_{0}\;\left|{f(n)\over g(n)}-1% \right|<\varepsilon
  197. \sim
  198. T ( n ) = 73 n 3 + 22 n 2 + 58 T(n)=73n^{3}+22n^{2}+58
  199. T ( n ) T(n)
  200. n n
  201. L n [ α , c ] = O ( e ( c + o ( 1 ) ) ( ln n ) α ( ln ln n ) 1 - α ) , L_{n}[\alpha,c]=O\left(e^{(c+o(1))(\ln n)^{\alpha}(\ln\ln n)^{1-\alpha}}\right),
  202. f g ( f - g ) o ( g ) f\sim g\iff(f-g)\in o(g)
  203. lim f / g = 1 \lim f/g=1
  204. Ω R \Omega_{R}
  205. Ω L \Omega_{L}
  206. Ω + \Omega_{+}
  207. Ω - \Omega_{-}
  208. f g f O ( g ) f\preceq g\iff f\in O(g)
  209. f g f o ( g ) ; f\prec g\iff f\in o(g);
  210. \prec\!\!\prec
  211. \ll
  212. \preceq
  213. \prec
  214. \ll
  215. O O
  216. f g f O ( g ) , f\ll g\iff f\in O(g),

Bijection.html

  1. \R + ( 0 , + ) \scriptstyle\R^{+}\;\equiv\;\left(0,\,+\infty\right)
  2. \R 0 + [ 0 , + ) \scriptstyle\R^{+}_{0}\;\equiv\;\left[0,\,+\infty\right)
  3. g f \scriptstyle g\,\circ\,f
  4. g f \scriptstyle g\,\circ\,f
  5. ( g f ) - 1 = ( f - 1 ) ( g - 1 ) \scriptstyle(g\,\circ\,f)^{-1}\;=\;(f^{-1})\,\circ\,(g^{-1})
  6. g f \scriptstyle g\,\circ\,f

Bilinear_transform.html

  1. H a ( s ) H_{a}(s)
  2. H d ( z ) H_{d}(z)
  3. j ω j\omega
  4. R e [ s ] = 0 Re[s]=0
  5. | z | = 1 |z|=1
  6. ( z - 1 ) \left(z^{-1}\right)
  7. H a ( j ω a ) H_{a}(j\omega_{a})
  8. H d ( e j ω d T ) H_{d}(e^{j\omega_{d}T})
  9. z = e s T = e s T / 2 e - s T / 2 1 + s T / 2 1 - s T / 2 \begin{aligned}\displaystyle z&\displaystyle=e^{sT}\\ &\displaystyle=\frac{e^{sT/2}}{e^{-sT/2}}\\ &\displaystyle\approx\frac{1+sT/2}{1-sT/2}\end{aligned}
  10. T T
  11. s s
  12. s = ( 1 / T ) ln ( z ) s=(1/T)\ln(z)\
  13. s = 1 T ln ( z ) = 2 T [ z - 1 z + 1 + 1 3 ( z - 1 z + 1 ) 3 + 1 5 ( z - 1 z + 1 ) 5 + 1 7 ( z - 1 z + 1 ) 7 + ] 2 T z - 1 z + 1 = 2 T 1 - z - 1 1 + z - 1 \begin{aligned}\displaystyle s&\displaystyle=\frac{1}{T}\ln(z)\\ &\displaystyle=\frac{2}{T}\left[\frac{z-1}{z+1}+\frac{1}{3}\left(\frac{z-1}{z+% 1}\right)^{3}+\frac{1}{5}\left(\frac{z-1}{z+1}\right)^{5}+\frac{1}{7}\left(% \frac{z-1}{z+1}\right)^{7}+\cdots\right]\\ &\displaystyle\approx\frac{2}{T}\frac{z-1}{z+1}\\ &\displaystyle=\frac{2}{T}\frac{1-z^{-1}}{1+z^{-1}}\end{aligned}
  14. H a ( s ) H_{a}(s)
  15. s 2 T z - 1 z + 1 . s\leftarrow\frac{2}{T}\frac{z-1}{z+1}.
  16. H d ( z ) = H a ( s ) | s = 2 T z - 1 z + 1 = H a ( 2 T z - 1 z + 1 ) . H_{d}(z)=H_{a}(s)\bigg|_{s=\frac{2}{T}\frac{z-1}{z+1}}=H_{a}\left(\frac{2}{T}% \frac{z-1}{z+1}\right).
  17. H a ( s ) = 1 / s C R + 1 / s C = 1 1 + R C s . \begin{aligned}\displaystyle H_{a}(s)&\displaystyle=\frac{1/sC}{R+1/sC}\\ &\displaystyle=\frac{1}{1+RCs}.\end{aligned}
  18. s s
  19. H d ( z ) H_{d}(z)
  20. = H a ( 2 T z - 1 z + 1 ) =H_{a}\left(\frac{2}{T}\frac{z-1}{z+1}\right)
  21. = 1 1 + R C ( 2 T z - 1 z + 1 ) =\frac{1}{1+RC\left(\frac{2}{T}\frac{z-1}{z+1}\right)}
  22. = 1 + z ( 1 - 2 R C / T ) + ( 1 + 2 R C / T ) z =\frac{1+z}{(1-2RC/T)+(1+2RC/T)z}
  23. = 1 + z - 1 ( 1 + 2 R C / T ) + ( 1 - 2 R C / T ) z - 1 . =\frac{1+z^{-1}}{(1+2RC/T)+(1-2RC/T)z^{-1}}.
  24. H a ( s ) = b 0 s 2 + b 1 s + b 2 a 0 s 2 + a 1 s + a 2 = b 0 + b 1 s - 1 + b 2 s - 2 a 0 + a 1 s - 1 + a 2 s - 2 H_{a}(s)=\frac{b_{0}s^{2}+b_{1}s+b_{2}}{a_{0}s^{2}+a_{1}s+a_{2}}=\frac{b_{0}+b% _{1}s^{-1}+b_{2}s^{-2}}{a_{0}+a_{1}s^{-1}+a_{2}s^{-2}}
  25. s K 1 - z - 1 1 + z - 1 s\leftarrow K\frac{1-z^{-1}}{1+z^{-1}}
  26. K 2 T K\triangleq\frac{2}{T}
  27. H d ( z ) = ( b 0 K 2 + b 1 K + b 2 ) + ( 2 b 2 - 2 b 0 K 2 ) z - 1 + ( b 0 K 2 - b 1 K + b 2 ) z - 2 ( a 0 K 2 + a 1 K + a 2 ) + ( 2 a 2 - 2 a 0 K 2 ) z - 1 + ( a 0 K 2 - a 1 K + a 2 ) z - 2 H_{d}(z)=\frac{(b_{0}K^{2}+b_{1}K+b_{2})+(2b_{2}-2b_{0}K^{2})z^{-1}+(b_{0}K^{2% }-b_{1}K+b_{2})z^{-2}}{(a_{0}K^{2}+a_{1}K+a_{2})+(2a_{2}-2a_{0}K^{2})z^{-1}+(a% _{0}K^{2}-a_{1}K+a_{2})z^{-2}}
  28. H d ( z ) = b 0 K 2 + b 1 K + b 2 a 0 K 2 + a 1 K + a 2 + 2 b 2 - 2 b 0 K 2 a 0 K 2 + a 1 K + a 2 z - 1 + b 0 K 2 - b 1 K + b 2 a 0 K 2 + a 1 K + a 2 z - 2 1 + 2 a 2 - 2 a 0 K 2 a 0 K 2 + a 1 K + a 2 z - 1 + a 0 K 2 - a 1 K + a 2 a 0 K 2 + a 1 K + a 2 z - 2 . H_{d}(z)=\frac{\frac{b_{0}K^{2}+b_{1}K+b_{2}}{a_{0}K^{2}+a_{1}K+a_{2}}+\frac{2% b_{2}-2b_{0}K^{2}}{a_{0}K^{2}+a_{1}K+a_{2}}z^{-1}+\frac{b_{0}K^{2}-b_{1}K+b_{2% }}{a_{0}K^{2}+a_{1}K+a_{2}}z^{-2}}{1+\frac{2a_{2}-2a_{0}K^{2}}{a_{0}K^{2}+a_{1% }K+a_{2}}z^{-1}+\frac{a_{0}K^{2}-a_{1}K+a_{2}}{a_{0}K^{2}+a_{1}K+a_{2}}z^{-2}}.
  29. y [ n ] = b 0 K 2 + b 1 K + b 2 a 0 K 2 + a 1 K + a 2 x [ n ] + 2 b 2 - 2 b 0 K 2 a 0 K 2 + a 1 K + a 2 x [ n - 1 ] + b 0 K 2 - b 1 K + b 2 a 0 K 2 + a 1 K + a 2 x [ n - 2 ] - 2 a 2 - 2 a 0 K 2 a 0 K 2 + a 1 K + a 2 y [ n - 1 ] - a 0 K 2 - a 1 K + a 2 a 0 K 2 + a 1 K + a 2 y [ n - 2 ] . \begin{aligned}\displaystyle y[n]=&\displaystyle\frac{b_{0}K^{2}+b_{1}K+b_{2}}% {a_{0}K^{2}+a_{1}K+a_{2}}\cdot x[n]+\frac{2b_{2}-2b_{0}K^{2}}{a_{0}K^{2}+a_{1}% K+a_{2}}\cdot x[n-1]\\ &\displaystyle{}+\frac{b_{0}K^{2}-b_{1}K+b_{2}}{a_{0}K^{2}+a_{1}K+a_{2}}\cdot x% [n-2]-\frac{2a_{2}-2a_{0}K^{2}}{a_{0}K^{2}+a_{1}K+a_{2}}\cdot y[n-1]\\ &\displaystyle{}-\frac{a_{0}K^{2}-a_{1}K+a_{2}}{a_{0}K^{2}+a_{1}K+a_{2}}\cdot y% [n-2].\end{aligned}
  30. H a ( s ) H_{a}(s)
  31. s = j ω s=j\omega
  32. j ω j\omega
  33. H d ( z ) H_{d}(z)
  34. z = e j ω T z=e^{j\omega T}
  35. | z | = 1 |z|=1
  36. ω \omega
  37. ω a \omega_{a}
  38. ω \omega
  39. H d ( z ) = H a ( 2 T z - 1 z + 1 ) H_{d}(z)=H_{a}\left(\frac{2}{T}\frac{z-1}{z+1}\right)
  40. H d ( e j ω T ) H_{d}(e^{j\omega T})
  41. = H a ( 2 T e j ω T - 1 e j ω T + 1 ) =H_{a}\left(\frac{2}{T}\frac{e^{j\omega T}-1}{e^{j\omega T}+1}\right)
  42. = H a ( 2 T e j ω T / 2 ( e j ω T / 2 - e - j ω T / 2 ) e j ω T / 2 ( e j ω T / 2 + e - j ω T / 2 ) ) =H_{a}\left(\frac{2}{T}\cdot\frac{e^{j\omega T/2}\left(e^{j\omega T/2}-e^{-j% \omega T/2}\right)}{e^{j\omega T/2}\left(e^{j\omega T/2}+e^{-j\omega T/2}% \right)}\right)
  43. = H a ( 2 T ( e j ω T / 2 - e - j ω T / 2 ) ( e j ω T / 2 + e - j ω T / 2 ) ) =H_{a}\left(\frac{2}{T}\cdot\frac{\left(e^{j\omega T/2}-e^{-j\omega T/2}\right% )}{\left(e^{j\omega T/2}+e^{-j\omega T/2}\right)}\right)
  44. = H a ( j 2 T ( e j ω T / 2 - e - j ω T / 2 ) / ( 2 j ) ( e j ω T / 2 + e - j ω T / 2 ) / 2 ) =H_{a}\left(j\frac{2}{T}\cdot\frac{\left(e^{j\omega T/2}-e^{-j\omega T/2}% \right)/(2j)}{\left(e^{j\omega T/2}+e^{-j\omega T/2}\right)/2}\right)
  45. = H a ( j 2 T sin ( ω T / 2 ) cos ( ω T / 2 ) ) =H_{a}\left(j\frac{2}{T}\cdot\frac{\sin(\omega T/2)}{\cos(\omega T/2)}\right)
  46. = H a ( j 2 T tan ( ω T / 2 ) ) =H_{a}\left(j\frac{2}{T}\cdot\tan\left(\omega T/2\right)\right)
  47. z = e j ω T z=e^{j\omega T}
  48. j ω j\omega
  49. s = j ω a s=j\omega_{a}
  50. ω a = 2 T tan ( ω T 2 ) \omega_{a}=\frac{2}{T}\tan\left(\omega\frac{T}{2}\right)
  51. ω = 2 T arctan ( ω a T 2 ) . \omega=\frac{2}{T}\arctan\left(\omega_{a}\frac{T}{2}\right).
  52. ω \omega
  53. ( 2 / T ) tan ( ω T / 2 ) (2/T)\tan(\omega T/2)
  54. ω \omega
  55. ( 2 / T ) tan ( ω T / 2 ) (2/T)\tan(\omega T/2)
  56. ω 2 / T \omega\ll 2/T
  57. ω a 2 / T \omega_{a}\ll 2/T
  58. ω ω a \omega\approx\omega_{a}
  59. - < ω a < + -\infty<\omega_{a}<+\infty
  60. - π T < ω < + π T . -\frac{\pi}{T}<\omega<+\frac{\pi}{T}.
  61. ω a = 0 \omega_{a}=0
  62. ω = 0 \omega=0
  63. ω a = ± \omega_{a}=\pm\infty
  64. ω = ± π / T . \omega=\pm\pi/T.
  65. ω a \omega_{a}
  66. ω . \omega.
  67. ω a = 2 T tan ( ω T 2 ) \omega_{a}=\frac{2}{T}\tan\left(\omega\frac{T}{2}\right)
  68. ω 0 \omega_{0}
  69. ω 0 0 \omega_{0}\to 0
  70. s ω 0 tan ( ω 0 T 2 ) z - 1 z + 1 . s\leftarrow\frac{\omega_{0}}{\tan(\frac{\omega_{0}T}{2})}\frac{z-1}{z+1}.

Binary_function.html

  1. f f
  2. X , Y , Z X,Y,Z
  3. f : X × Y Z \,f\colon X\times Y\rightarrow Z
  4. X × Y X\times Y
  5. X X
  6. Y . Y.
  7. X Y X\otimes Y

Binary_heap.html

  1. O ( n log n ) O(n\log n)
  2. O ( log n ) O(\log n)
  3. n n
  4. h h
  5. h + 1 h+1
  6. O ( h ) O(h)
  7. log ( n ) \left\lfloor\log(n)\right\rfloor
  8. h h
  9. 2 ( log n - h ) - 1 = 2 log n 2 h + 1 = n 2 h + 1 \leq\left\lceil 2^{\left(\log n-h\right)-1}\right\rceil=\left\lceil\frac{2^{% \log n}}{2^{h+1}}\right\rceil=\left\lceil\frac{n}{2^{h+1}}\right\rceil
  10. h = 0 log n n 2 h + 1 O ( h ) \displaystyle\sum_{h=0}^{\lceil\log n\rceil}\frac{n}{2^{h+1}}O(h)
  11. 2 n - 2 s 2 ( n ) - e 2 ( n ) 2n-2s_{2}(n)-e_{2}(n)
  12. O O
  13. ( log n ) (\log n)
  14. i i
  15. right = 2 i + 2 \,\text{right}=2i+2
  16. i i
  17. L L
  18. l l
  19. 2 l 2^{l}
  20. 2 l + 1 - 1 2^{l+1}-1
  21. l l
  22. k k
  23. ( k - 1 ) (k-1)
  24. last ( l ) = ( 2 l + 1 - 1 ) - 1 = 2 l + 1 - 2 \,\text{last}(l)=(2^{l+1}-1)-1=2^{l+1}-2
  25. j j
  26. i i
  27. i = \displaystyle i=
  28. j j
  29. 2 j 2j
  30. i i
  31. L + 1 L+1
  32. right = \displaystyle\,\text{right}=
  33. left = 2 i + 1 \,\text{left}=2i+1
  34. 2 l + 1 - 1 2^{l+1}-1
  35. left = 2 i \,\text{left}=2i
  36. right = 2 i + 1 \,\text{right}=2i+1
  37. i = 2 × ( parent ) + 1 i=2\times(\,\text{parent})+1
  38. i = 2 × ( parent ) + 2 i=2\times(\,\text{parent})+2
  39. parent = i - 1 2 or i - 2 2 \,\text{parent}=\frac{i-1}{2}\,\textbf{ or }\frac{i-2}{2}
  40. i - 1 2 \left\lfloor\dfrac{i-1}{2}\right\rfloor
  41. i i
  42. i i
  43. ( i - 2 ) (i-2)
  44. ( i - 1 ) (i-1)
  45. i - 1 2 = \displaystyle\left\lfloor\dfrac{i-1}{2}\right\rfloor=
  46. parent = i - 1 2 \,\text{parent}=\left\lfloor\dfrac{i-1}{2}\right\rfloor

Binary_operation.html

  1. f : S × S S . \,f\colon S\times S\rightarrow S.

Binary_relation.html

  1. G = { ( 1 , 2 ) , ( 1 , 3 ) , ( 2 , 7 ) } G=\{(1,2),(1,3),(2,7)\}
  2. ( , , G ) (\mathbb{Z},\mathbb{Z},G)
  3. ( , , G ) (\mathbb{R},\mathbb{N},G)
  4. ( , , G ) (\mathbb{N},\mathbb{R},G)
  5. \mathbb{Z}
  6. \mathbb{R}
  7. R R
  8. x x
  9. y y
  10. ( x , y ) R (x,y)\in R
  11. R R
  12. y y
  13. x x
  14. ( x , y ) R (x,y)\in R
  15. R R
  16. f : f:\mathbb{R}\rightarrow\mathbb{R}
  17. f : + f:\mathbb{R}\rightarrow\mathbb{R}^{+}

Binary_search_algorithm.html

  1. log 2 N \lfloor\log_{2}N\rfloor
  2. 2 log 2 k + 1 2\lfloor\log_{2}k\rfloor+1
  3. 2 log 2 k 2\lceil\log_{2}k\rceil
  4. log 2 k \log_{2}k
  5. m n n ! m^{n}n!
  6. O ( l o g N ) O(logN)
  7. N N
  8. N / 2 N/2
  9. N / 4 N/4
  10. 3 N / 4 3N/4
  11. l o g ( n ) log(n)
  12. p p
  13. 2 log 2 n 2\lceil\log_{2}n\rceil
  14. n n
  15. p p

Binary_search_tree.html

  1. n √n
  2. l o g n logn
  3. O ( n 2 ) O(n^{2})

Binary_star.html

  1. r 1 = a m 2 m 1 + m 2 = a 1 + m 1 / m 2 r_{1}=a\cdot{m_{2}\over m_{1}+m_{2}}={a\over 1+m_{1}/m_{2}}

Binary_tree.html

  1. n n
  2. n = 2 ( h + 1 ) - 1 n=2(h+1)-1
  3. n = 2 h + 1 - 1 n=2^{h+1}-1
  4. h h
  5. l l
  6. l = ( n + 1 ) / 2 l=(n+1)/2
  7. n - l = k = 0 log 2 ( l ) - 1 2 k = 2 log 2 ( l ) - 1 = l - 1 n-l=\sum_{k=0}^{\log_{2}(l)-1}2^{k}=2^{\log_{2}(l)}-1=l-1
  8. l l
  9. n = 2 l - 1 n=2l-1
  10. h = log 2 ( l ) + 1 = log 2 ( ( n + 1 ) / 2 ) + 1 = log 2 ( n + 1 ) h=\lceil\log_{2}(l)\rceil+1=\lceil\log_{2}((n+1)/2)\rceil+1=\lceil\log_{2}(n+1)\rceil
  11. l = 2 h l=2^{h}
  12. n = 2 h + 1 - 1 n=2^{h+1}-1
  13. ( ( X * X ) * X ) * X , ( X * ( X * X ) ) * X , ( X * X ) * ( X * X ) , X * ( ( X * X ) * X ) , X * ( X * ( X * X ) ) . ((X*X)*X)*X,\qquad(X*(X*X))*X,\qquad(X*X)*(X*X),\qquad X*((X*X)*X),\qquad X*(X% *(X*X)).
  14. C n C_{n}
  15. C 0 = 1 C_{0}=1
  16. C n = i = 0 n - 1 C i C n - 1 - i \textstyle C_{n}=\sum_{i=0}^{n-1}C_{i}C_{n-1-i}
  17. C n C_{n}
  18. C n C_{n}
  19. ( ) ( ) ( ) , ( ) ( ( ) ) , ( ( ) ) ( ) , ( ( ) ( ) ) , ( ( ( ) ) ) ()()(),\qquad()(()),\qquad(())(),\qquad(()()),\qquad((()))
  20. 2 i + 1 2i+1
  21. 2 i + 2 2i+2
  22. i - 1 2 \left\lfloor\frac{i-1}{2}\right\rfloor
  23. n n
  24. C n \mathrm{C}_{n}
  25. n n
  26. n n
  27. 4 n 4^{n}
  28. log 2 4 n = 2 n \log_{2}4^{n}=2n
  29. 2 n + o ( n ) 2n+o(n)
  30. 2 n + 1 2n+1
  31. n n

Binomial_coefficient.html

  1. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}
  2. n ! k ! ( n - k ) ! \tfrac{n!}{k!\,(n-k)!}
  3. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}
  4. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}
  5. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}
  6. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}
  7. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}
  8. ( x + y ) n = k = 0 n ( n k ) x n - k y k (x+y)^{n}=\sum_{k=0}^{n}{\left({{n}\atop{k}}\right)}x^{n-k}y^{k}
  9. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}
  10. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}
  11. k = a 1 + a 2 + + a n k=a_{1}+a_{2}+\cdots+a_{n}
  12. ( n + k - 1 n - 1 ) {\textstyle\left({{n+k-1}\atop{n-1}}\right)}
  13. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}
  14. ( n k ) = ( n - 1 k - 1 ) + ( n - 1 k ) for all integers n , k : 1 k n - 1 , {\left({{n}\atop{k}}\right)}={\left({{n-1}\atop{k-1}}\right)}+{\left({{n-1}% \atop{k}}\right)}\quad\,\text{for all integers }n,k:1\leq k\leq n-1,
  15. ( n 0 ) = ( n n ) = 1 for all integers n 0 , {\left({{n}\atop{0}}\right)}={\left({{n}\atop{n}}\right)}=1\quad\,\text{for % all integers }n\geq 0,
  16. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}
  17. n k ¯ n^{\underline{k}}
  18. ( n k ) = n ! k ! ( n - k ) ! for 0 k n , {\left({{n}\atop{k}}\right)}=\frac{n!}{k!\,(n-k)!}\quad\,\text{for }\ 0\leq k% \leq n,
  19. ( n k ) = ( n n - k ) for 0 k n , {\left({{n}\atop{k}}\right)}={\left({{n}\atop{n-k}}\right)}\quad\,\text{for }% \ 0\leq k\leq n,
  20. ( n k ) = { n k ¯ / k ! if k n 2 n n - k ¯ / ( n - k ) ! if k > n 2 . {\left({{n}\atop{k}}\right)}=\begin{cases}n^{\underline{k}}/k!&\,\text{if }\ k% \leq\frac{n}{2}\\ n^{\underline{n-k}}/(n-k)!&\,\text{if }\ k>\frac{n}{2}\end{cases}.
  21. ( α k ) = α k ¯ k ! = α ( α - 1 ) ( α - 2 ) ( α - k + 1 ) k ( k - 1 ) ( k - 2 ) 1 for k 𝒩 and arbitrary α . {\left({{\alpha}\atop{k}}\right)}=\frac{\alpha^{\underline{k}}}{k!}=\frac{% \alpha(\alpha-1)(\alpha-2)\cdots(\alpha-k+1)}{k(k-1)(k-2)\cdots 1}\quad\,\text% {for }k\in\mathcal{N}\,\text{ and arbitrary }\alpha.
  22. ( α k ) {\textstyle\left({{\alpha}\atop{k}}\right)}
  23. ( 1 + X ) α = k = 0 ( α k ) X k . (1+X)^{\alpha}=\sum_{k=0}^{\infty}{\alpha\choose k}X^{k}.
  24. ( 1 + X ) α ( 1 + X ) β = ( 1 + X ) α + β and ( ( 1 + X ) α ) β = ( 1 + X ) α β . (1+X)^{\alpha}(1+X)^{\beta}=(1+X)^{\alpha+\beta}\quad\,\text{and}\quad((1+X)^{% \alpha})^{\beta}=(1+X)^{\alpha\beta}.
  25. ( n k ) + ( n k + 1 ) = ( n + 1 k + 1 ) , {n\choose k}+{n\choose k+1}={n+1\choose k+1},
  26. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}
  27. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}
  28. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}
  29. ( n + k - 1 k ) {\textstyle\left({{n+k-1}\atop{k}}\right)}
  30. ( n + k k ) {\textstyle\left({{n+k}\atop{k}}\right)}
  31. ( n + 1 k ) {\textstyle\left({{n+1}\atop{k}}\right)}
  32. 1 n + 1 ( 2 n n ) . \tfrac{1}{n+1}{\textstyle\left({{2n}\atop{n}}\right)}.
  33. ( n k ) p k ( 1 - p ) n - k . {\textstyle\left({{n}\atop{k}}\right)}p^{k}(1-p)^{n-k}\!.
  34. ( t k ) \scriptstyle{{\left({{t}\atop{k}}\right)}}
  35. ( t k ) = ( t ) k k ! = ( t ) k ( k ) k = t ( t - 1 ) ( t - 2 ) ( t - k + 1 ) k ( k - 1 ) ( k - 2 ) 2 1 ; {\left({{t}\atop{k}}\right)}=\frac{(t)_{k}}{k!}=\frac{(t)_{k}}{(k)_{k}}=\frac{% t(t-1)(t-2)\cdots(t-k+1)}{k(k-1)(k-2)\cdots 2\cdot 1};\,\!
  36. ( t k ) {\textstyle\left({{t}\atop{k}}\right)}
  37. ( t k ) = i = 0 k [ k i ] t i k ! . {\left({{t}\atop{k}}\right)}=\sum_{i=0}^{k}\left[{k\atop i}\right]\frac{t^{i}}% {k!}.
  38. ( t k ) {\textstyle\left({{t}\atop{k}}\right)}
  39. d d t ( t k ) = ( t k ) i = 0 k - 1 1 t - i . \frac{\mathrm{d}}{\mathrm{d}t}{\left({{t}\atop{k}}\right)}={\left({{t}\atop{k}% }\right)}\sum_{i=0}^{k-1}\frac{1}{t-i}\,.
  40. k = 0 d a k ( t k ) \sum_{k=0}^{d}a_{k}{\left({{t}\atop{k}}\right)}
  41. a k = i = 0 k ( - 1 ) k - i ( k i ) p ( i ) . a_{k}=\sum_{i=0}^{k}(-1)^{k-i}{\left({{k}\atop{i}}\right)}p(i).
  42. ( t k ) {\textstyle\left({{t}\atop{k}}\right)}
  43. 9 ( t 2 ) + 6 ( t 1 ) + 0 ( t 0 ) . 9{\textstyle\left({{t}\atop{2}}\right)}+6{\textstyle\left({{t}\atop{1}}\right)% }+0{\textstyle\left({{t}\atop{0}}\right)}.
  44. ( n k ) = n k ( n - 1 k - 1 ) {\left({{n}\atop{k}}\right)}=\frac{n}{k}{\left({{n-1}\atop{k-1}}\right)}
  45. ( n - 1 k ) - ( n - 1 k - 1 ) = n - 2 k n ( n k ) . {\left({{n-1}\atop{k}}\right)}-{\left({{n-1}\atop{k-1}}\right)}=\frac{n-2k}{n}% {\left({{n}\atop{k}}\right)}.
  46. ( n h ) ( n - h k ) = ( n k ) ( n - k h ) . {\left({{n}\atop{h}}\right)}{\left({{n-h}\atop{k}}\right)}={\left({{n}\atop{k}% }\right)}{\left({{n-k}\atop{h}}\right)}.
  47. ( n k ) = n + 1 - k k ( n k - 1 ) . {\left({{n}\atop{k}}\right)}=\frac{n+1-k}{k}{\left({{n}\atop{k-1}}\right)}.
  48. k = 0 n ( n k ) = 2 n \sum_{k=0}^{n}{\textstyle\left({{n}\atop{k}}\right)}=2^{n}
  49. k = 0 n k ( n k ) = n 2 n - 1 \sum_{k=0}^{n}k{\textstyle\left({{n}\atop{k}}\right)}=n2^{n-1}
  50. k = 0 n k 2 ( n k ) = ( n + n 2 ) 2 n - 2 \sum_{k=0}^{n}k^{2}{\textstyle\left({{n}\atop{k}}\right)}=(n+n^{2})2^{n-2}
  51. j = 0 k ( m j ) ( n - m k - j ) = ( n k ) \sum_{j=0}^{k}{\textstyle\left({{m}\atop{j}}\right)}{\textstyle\left({{n-m}% \atop{k-j}}\right)}={\textstyle\left({{n}\atop{k}}\right)}
  52. x k x^{k}
  53. m = 0 n ( m j ) ( n - m k - j ) = ( n + 1 k + 1 ) , \sum_{m=0}^{n}{\textstyle\left({{m}\atop{j}}\right)}{\textstyle\left({{n-m}% \atop{k-j}}\right)}={\textstyle\left({{n+1}\atop{k+1}}\right)}\,,
  54. x n + 1 x^{n+1}
  55. x ( x j ( 1 - x ) j + 1 ) ( x k - j ( 1 - x ) k - j + 1 ) = x k + 1 ( 1 - x ) k + 2 x\left(\tfrac{x^{j}}{(1-x)^{j+1}}\right)\left(\tfrac{x^{k-j}}{(1-x)^{k-j+1}}% \right)=\tfrac{x^{k+1}}{(1-x)^{k+2}}
  56. x l ( 1 - x ) l + 1 = p = 0 ( p l ) x p . \tfrac{x^{l}}{(1-x)^{l+1}}=\sum_{p=0}^{\infty}{\textstyle\left({{p}\atop{l}}% \right)}x^{p}\,.
  57. m = 0 n ( m k ) = ( n + 1 k + 1 ) . \sum_{m=0}^{n}{\textstyle\left({{m}\atop{k}}\right)}={\textstyle\left({{n+1}% \atop{k+1}}\right)}\,.
  58. j = 0 m ( m j ) 2 = ( 2 m m ) . \sum_{j=0}^{m}{\textstyle\left({{m}\atop{j}}\right)}^{2}={\textstyle\left({{2m% }\atop{m}}\right)}.
  59. k = 0 n 2 ( n - k k ) = F ( n + 1 ) . \sum_{k=0}^{\lfloor\frac{n}{2}\rfloor}{\textstyle\left({{n-k}\atop{k}}\right)}% =F(n+1).
  60. j = k n ( n + 1 - j ) ( j - 1 k - 1 ) = ( n + 1 k + 1 ) . \sum_{j=k}^{n}(n+1-j){\textstyle\left({{j-1}\atop{k-1}}\right)}={\textstyle% \left({{n+1}\atop{k+1}}\right)}.
  61. j = 0 k ( n j ) \sum_{j=0}^{k}{\textstyle\left({{n}\atop{j}}\right)}
  62. j = 0 k ( - 1 ) j ( n j ) = ( - 1 ) k ( n - 1 k ) \sum_{j=0}^{k}(-1)^{j}{\textstyle\left({{n}\atop{j}}\right)}=(-1)^{k}{% \textstyle\left({{n-1}\atop{k}}\right)}
  63. j = 0 n ( - 1 ) j ( n j ) = 0 \sum_{j=0}^{n}(-1)^{j}{\textstyle\left({{n}\atop{j}}\right)}=0
  64. j = 0 n ( - 1 ) j ( n j ) P ( j ) = 0. \sum_{j=0}^{n}(-1)^{j}{\textstyle\left({{n}\atop{j}}\right)}P(j)=0.
  65. P ( x ) = x ( x - 1 ) ( x - k + 1 ) P(x)=x(x-1)\cdots(x-k+1)
  66. a n a_{n}
  67. j = 0 n ( - 1 ) j ( n j ) P ( m + ( n - j ) d ) = d n n ! a n \sum_{j=0}^{n}(-1)^{j}{\textstyle\left({{n}\atop{j}}\right)}P(m+(n-j)d)=d^{n}n% !a_{n}
  68. k - 1 k j = 0 1 ( j + x k ) = 1 ( x - 1 k - 1 ) \frac{k-1}{k}\sum_{j=0}^{\infty}\frac{1}{{\left({{j+x}\atop{k}}\right)}}=\frac% {1}{{\left({{x-1}\atop{k-1}}\right)}}
  69. k - 1 k j = 0 M 1 ( j + x k ) = 1 ( x - 1 k - 1 ) - 1 ( M + x k - 1 ) \frac{k-1}{k}\sum_{j=0}^{M}\frac{1}{{\left({{j+x}\atop{k}}\right)}}=\frac{1}{{% \left({{x-1}\atop{k-1}}\right)}}-\frac{1}{{\left({{M+x}\atop{k-1}}\right)}}
  70. i = 0 n i ( n i ) 2 = n 2 ( 2 n n ) \sum_{i=0}^{n}{i{\left({{n}\atop{i}}\right)}^{2}}=\frac{n}{2}{\left({{2n}\atop% {n}}\right)}
  71. i = 0 n i 2 ( n i ) 2 = n 2 ( 2 n - 2 n - 1 ) . \sum_{i=0}^{n}{i^{2}{\left({{n}\atop{i}}\right)}^{2}}=n^{2}{\left({{2n-2}\atop% {n-1}}\right)}.
  72. ( 0 t < s ) (0\leqslant t<s)
  73. ( n t ) + ( n t + s ) + ( n t + 2 s ) + = 1 s j = 0 s - 1 ( 2 cos π j s ) n cos π ( n - 2 t ) j s . {\left({{n}\atop{t}}\right)}+{\left({{n}\atop{t+s}}\right)}+{\left({{n}\atop{t% +2s}}\right)}+\ldots=\frac{1}{s}\sum_{j=0}^{s-1}\left(2\cos\frac{\pi j}{s}% \right)^{n}\cos\frac{\pi(n-2t)j}{s}.
  74. n q {n}\geq{q}
  75. k = q n ( n k ) ( k q ) = 2 n - q ( n q ) \sum_{k=q}^{n}{\textstyle\left({{n}\atop{k}}\right)}{\textstyle\left({{k}\atop% {q}}\right)}=2^{n-q}{\textstyle\left({{n}\atop{q}}\right)}
  76. ( n q ) {\textstyle\left({{n}\atop{q}}\right)}
  77. 2 n - q . 2^{n-q}.
  78. ( n k ) = ( n - 1 k - 1 ) + ( n - 1 k ) {n\choose k}={n-1\choose k-1}+{n-1\choose k}
  79. k = 0 n ( n k ) 2 = ( 2 n n ) . \sum_{k=0}^{n}{\textstyle\left({{n}\atop{k}}\right)}^{2}={\textstyle\left({{2n% }\atop{n}}\right)}.
  80. 2 n 2n
  81. ( 2 n n ) {\textstyle\left({{2n}\atop{n}}\right)}
  82. n - k n-k
  83. k = 0 n ( n k ) ( n n - k ) = ( 2 n n ) . \sum_{k=0}^{n}{\textstyle\left({{n}\atop{k}}\right)}{\textstyle\left({{n}\atop% {n-k}}\right)}={\textstyle\left({{2n}\atop{n}}\right)}.
  84. k = 0 n 2 ( n - k k ) = F ( n + 1 ) \sum_{k=0}^{\lfloor\frac{n}{2}\rfloor}{\textstyle\left({{n-k}\atop{k}}\right)}% =F(n+1)
  85. ( n - k k ) {\textstyle\left({{n-k}\atop{k}}\right)}
  86. ( 0 , 0 ) (0,0)
  87. ( k , n - k ) (k,n-k)
  88. ( 0 , 1 ) (0,1)
  89. ( 1 , 1 ) (1,1)
  90. ( n - k ) (n-k)
  91. k k
  92. ( 1 , 1 ) (1,1)
  93. ( 1 , 1 ) (1,1)
  94. ( 0 , 2 ) (0,2)
  95. k k
  96. ( 0 , n ) (0,n)
  97. ( 0 , 1 ) (0,1)
  98. ( 0 , 2 ) (0,2)
  99. k k
  100. 0
  101. n 2 \lfloor\frac{n}{2}\rfloor
  102. ( 0 , 0 ) (0,0)
  103. ( 0 , n ) (0,n)
  104. ( 0 , 1 ) (0,1)
  105. ( 0 , 2 ) (0,2)
  106. F ( n + 1 ) F(n+1)
  107. 0 k n ( n k ) = 2 n \sum_{0\leq{k}\leq{n}}{\left({{n}\atop{k}}\right)}=2^{n}
  108. 2 n - 1 2^{n}-1
  109. k = - a a ( - 1 ) k ( 2 a k + a ) 3 = ( 3 a ) ! ( a ! ) 3 \sum_{k=-a}^{a}(-1)^{k}{2a\choose k+a}^{3}=\frac{(3a)!}{(a!)^{3}}
  110. k = - a a ( - 1 ) k ( a + b a + k ) ( b + c b + k ) ( c + a c + k ) = ( a + b + c ) ! a ! b ! c ! , \sum_{k=-a}^{a}(-1)^{k}{a+b\choose a+k}{b+c\choose b+k}{c+a\choose c+k}=\frac{% (a+b+c)!}{a!\,b!\,c!}\,,
  111. m , n \textstyle m,n\in\mathbb{N}
  112. - π π cos ( ( 2 m - n ) x ) cos n x d x = π 2 n - 1 ( n m ) \int_{-\pi}^{\pi}\cos((2m-n)x)\cos^{n}x\ dx=\frac{\pi}{2^{n-1}}{\left({{n}% \atop{m}}\right)}
  113. - π π sin ( ( 2 m - n ) x ) sin n x d x = { ( - 1 ) m + ( n + 1 ) / 2 π 2 n - 1 ( n m ) n odd 0 otherwise \int_{-\pi}^{\pi}\sin((2m-n)x)\sin^{n}x\ dx=\left\{\begin{array}[]{cc}(-1)^{m+% (n+1)/2}\frac{\pi}{2^{n-1}}{\left({{n}\atop{m}}\right)}&n\,\text{ odd}\\ 0&\,\text{otherwise}\\ \end{array}\right.
  114. - π π cos ( ( 2 m - n ) x ) sin n x d x = { ( - 1 ) m + ( n + 1 ) / 2 π 2 n - 1 ( n m ) n even 0 otherwise \int_{-\pi}^{\pi}\cos((2m-n)x)\sin^{n}x\ dx=\left\{\begin{array}[]{cc}(-1)^{m+% (n+1)/2}\frac{\pi}{2^{n-1}}{\left({{n}\atop{m}}\right)}&n\,\text{ even}\\ 0&\,\text{otherwise}\\ \end{array}\right.
  115. ( n 0 ) , ( n 1 ) , ( n 2 ) , {n\choose 0},\;{n\choose 1},\;{n\choose 2},\;\ldots
  116. k ( n k ) x k = ( 1 + x ) n . \sum_{k}{n\choose k}x^{k}=(1+x)^{n}.
  117. ( 0 k ) , ( 1 k ) , ( 2 k ) , {0\choose k},\;{1\choose k},\;{2\choose k},\;\ldots
  118. n = k ( n k ) y n = y k ( 1 - y ) k + 1 . \sum_{n=k}^{\infty}{n\choose k}y^{n}=\frac{y^{k}}{(1-y)^{k+1}}.
  119. n , k ( n k ) x k y n = 1 1 - y - x y . \sum_{n,k}{n\choose k}x^{k}y^{n}=\frac{1}{1-y-xy}.
  120. n , k ( n + k k ) x k y n = 1 1 - x - y . \sum_{n,k}{n+k\choose k}x^{k}y^{n}=\frac{1}{1-x-y}.
  121. n , k 1 ( n + k ) ! ( n + k k ) x k y n = e x + y . \sum_{n,k}\frac{1}{(n+k)!}{n+k\choose k}x^{k}y^{n}=e^{x+y}.
  122. ( m + n m ) {\textstyle\left({{m+n}\atop{m}}\right)}
  123. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}
  124. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}
  125. ( p r s ) {\textstyle\left({{p^{r}}\atop{s}}\right)}
  126. ( 9 6 ) {\textstyle\left({{9}\atop{6}}\right)}
  127. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}
  128. lim N f ( N ) N ( N + 1 ) / 2 = 1. \lim_{N\to\infty}\frac{f(N)}{N(N+1)/2}=1.
  129. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}
  130. ( p k ) = p ( p - 1 ) ( p - k + 1 ) k ( k - 1 ) 1 {\left({{p}\atop{k}}\right)}=\frac{p\cdot(p-1)\cdots(p-k+1)}{k\cdot(k-1)\cdots 1}
  131. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}
  132. ( n k ) k ( n k ) n k k ! ( n e k ) k \left(\frac{n}{k}\right)^{k}\leq{n\choose k}\leq\frac{n^{k}}{k!}\leq\left(% \frac{n\cdot e}{k}\right)^{k}
  133. n ( 2 n n ) 2 2 n - 1 \sqrt{n}{2n\choose n}\geq 2^{2n-1}
  134. n ( m n n ) m m ( n - 1 ) + 1 ( m - 1 ) ( m - 1 ) ( n - 1 ) \sqrt{n}{mn\choose n}\geq\frac{m^{m(n-1)+1}}{(m-1)^{(m-1)(n-1)}}
  135. ( 2 n n ) 4 n π n {2n\choose n}\sim\frac{4^{n}}{\sqrt{\pi n}}
  136. n . n\rightarrow\infty\,.
  137. n n
  138. k k
  139. log 2 ( n k ) n H ( k n ) \log_{2}{n\choose k}\sim nH\left(\frac{k}{n}\right)
  140. H ( ϵ ) = - ϵ log 2 ( ϵ ) - ( 1 - ϵ ) log 2 ( 1 - ϵ ) H(\epsilon)=-\epsilon\log_{2}(\epsilon)-(1-\epsilon)\log_{2}(1-\epsilon)
  141. ϵ \epsilon
  142. n k 1 n\geq k\geq 1
  143. ϵ k / n 1 / 2 \epsilon\doteq k/n\leq 1/2
  144. k + 1 k+1
  145. 1 8 n ϵ ( 1 - ϵ ) 2 H ( ϵ ) n i = 0 k ( n i ) 2 H ( ϵ ) n . \frac{1}{\sqrt{8n\epsilon(1-\epsilon)}}\cdot 2^{H(\epsilon)\cdot n}\leq\sum_{i% =0}^{k}{\left({{n}\atop{i}}\right)}\leq 2^{H(\epsilon)\cdot n}\,.
  146. n n
  147. k k
  148. n n
  149. ( n k ) = n ( n - 1 ) ( n - k + 1 ) k ! ( n - k / 2 ) k k k e - k 2 π k = ( n / k - 0.5 ) k e k 2 π k , {n\choose k}=\frac{n(n-1)\dots(n-k+1)}{k!}\approx\frac{(n-k/2)^{k}}{k^{k}e^{-k% }\sqrt{2\pi k}}=\frac{(n/k-0.5)^{k}e^{k}}{\sqrt{2\pi k}}\,,
  150. log ( n k ) k ln ( n / k - 0.5 ) + k - 0.5 ln ( 2 π k ) . \log{n\choose k}\approx k\ln(n/k-0.5)+k-0.5\ln(2\pi k)\,.
  151. ln ( n ( n - 1 ) ( n - k + 1 ) ) \ln{(n(n-1)\dots(n-k+1))}
  152. log ( n k ) ( n + 0.5 ) ln n + 0.5 n - k + 0.5 + k ln n - k + 0.5 k - 0.5 ln ( 2 π k ) \log{n\choose k}\approx(n+0.5)\ln\frac{n+0.5}{n-k+0.5}+k\ln\frac{n-k+0.5}{k}-0% .5\ln(2\pi k)
  153. n = 20 n=20
  154. k = 10 k=10
  155. log ( n k ) 12.127 \log{{\textstyle\left({{n}\atop{k}}\right)}}\approx 12.127
  156. ( - 1 ) k ( z k ) = ( - z + k - 1 k ) = 1 Γ ( - z ) 1 ( k + 1 ) z + 1 j = k + 1 ( 1 + 1 j ) - z - 1 1 - z + 1 j (-1)^{k}{z\choose k}={-z+k-1\choose k}=\frac{1}{\Gamma(-z)}\frac{1}{(k+1)^{z+1% }}\prod_{j=k+1}\frac{(1+\frac{1}{j})^{-z-1}}{1-\frac{z+1}{j}}
  157. ( z k ) ( - 1 ) k Γ ( - z ) k z + 1 and ( z + k k ) = k z Γ ( z + 1 ) ( 1 + z ( z + 1 ) 2 k + 𝒪 ( k - 2 ) ) {z\choose k}\approx\frac{(-1)^{k}}{\Gamma(-z)k^{z+1}}\qquad\mathrm{and}\qquad{% z+k\choose k}=\frac{k^{z}}{\Gamma(z+1)}\left(1+\frac{z(z+1)}{2k}+\mathcal{O}% \left(k^{-2}\right)\right)
  158. k k\to\infty
  159. ( z + k k ) e z ( H k - γ ) Γ ( z + 1 ) {z+k\choose k}\approx\frac{e^{z(H_{k}-\gamma)}}{\Gamma(z+1)}
  160. H k H_{k}
  161. γ \gamma
  162. < m t p l > ( z + k j ) \frac{<}{m}tpl>{{z+k\choose j}}
  163. k k\to\infty
  164. j / k x j/k\to x
  165. x x
  166. i = 0 k ( n i ) i = 0 k n i 1 k - i = ( 1 + n ) k \sum_{i=0}^{k}{n\choose i}\leq\sum_{i=0}^{k}n^{i}\cdot 1^{k-i}=(1+n)^{k}
  167. ( n k ) = 2 n 1 2 n π e - ( k - ( n / 2 ) ) 2 n / 2 [ 1 + O ( 1 n ) ] . {\left({{n}\atop{k}}\right)}=\frac{2^{n}}{\sqrt{\frac{1}{2}n\pi}}e^{-\frac{(k-% (n/2))^{2}}{n/2}}\left[1+O\left(\frac{1}{\sqrt{n}}\right)\right].
  168. n p , n p ( 1 - p ) np,np(1-p)
  169. p = 1 - p = 1 / 2 p=1-p=1/2
  170. ( n k 1 , k 2 , , k r ) = n ! k 1 ! k 2 ! k r ! {n\choose k_{1},k_{2},\ldots,k_{r}}=\frac{n!}{k_{1}!k_{2}!\cdots k_{r}!}
  171. i = 1 r k i = n . \sum_{i=1}^{r}k_{i}=n.
  172. ( x 1 + x 2 + + x r ) n . (x_{1}+x_{2}+\cdots+x_{r})^{n}.
  173. ( n k 1 , k 2 ) = ( n k 1 , n - k 1 ) = ( n k 1 ) = ( n k 2 ) . {n\choose k_{1},k_{2}}={n\choose k_{1},n-k_{1}}={n\choose k_{1}}={n\choose k_{% 2}}.
  174. ( n k 1 , k 2 , , k r ) = ( n - 1 k 1 - 1 , k 2 , , k r ) + ( n - 1 k 1 , k 2 - 1 , , k r ) + + ( n - 1 k 1 , k 2 , , k r - 1 ) {n\choose k_{1},k_{2},\ldots,k_{r}}={n-1\choose k_{1}-1,k_{2},\ldots,k_{r}}+{n% -1\choose k_{1},k_{2}-1,\ldots,k_{r}}+\ldots+{n-1\choose k_{1},k_{2},\ldots,k_% {r}-1}
  175. ( n k 1 , k 2 , , k r ) = ( n k σ 1 , k σ 2 , , k σ r ) {n\choose k_{1},k_{2},\ldots,k_{r}}={n\choose k_{\sigma_{1}},k_{\sigma_{2}},% \ldots,k_{\sigma_{r}}}
  176. ( σ i ) (\sigma_{i})
  177. z 0 z_{0}
  178. ( z k ) = 1 k ! i = 0 k z i s k , i = i = 0 k ( z - z 0 ) i j = i k ( z 0 j - i ) s k + i - j , i ( k + i - j ) ! = i = 0 k ( z - z 0 ) i j = i k z 0 j - i ( j i ) s k , j k ! . \begin{aligned}\displaystyle{z\choose k}=\frac{1}{k!}\sum_{i=0}^{k}z^{i}s_{k,i% }&\displaystyle=\sum_{i=0}^{k}(z-z_{0})^{i}\sum_{j=i}^{k}{z_{0}\choose j-i}% \frac{s_{k+i-j,i}}{(k+i-j)!}\\ &\displaystyle=\sum_{i=0}^{k}(z-z_{0})^{i}\sum_{j=i}^{k}z_{0}^{j-i}{j\choose i% }\frac{s_{k,j}}{k!}.\end{aligned}
  179. n n
  180. k k
  181. k k
  182. ( 1 / 2 k ) = ( 2 k k ) ( - 1 ) k + 1 2 2 k ( 2 k - 1 ) . {{1/2}\choose{k}}={{2k}\choose{k}}\frac{(-1)^{k+1}}{2^{2k}(2k-1)}.
  183. 1 + x \sqrt{1+x}
  184. 1 + x = k 0 ( 1 / 2 k ) x k . \sqrt{1+x}=\sum_{k\geqslant 0}{{\textstyle\left({{1/2}\atop{k}}\right)}}x^{k}.
  185. ( z m ) ( z n ) = k = 0 m ( m + n - k k , m - k , n - k ) ( z m + n - k ) {z\choose m}{z\choose n}=\sum_{k=0}^{m}{m+n-k\choose k,m-k,n-k}{z\choose m+n-k}
  186. 1 < m t p l > ( z n ) \frac{1}{<}mtpl>{{z\choose n}}
  187. 1 < m t p l > ( z + n n ) \frac{1}{<}mtpl>{{z+n\choose n}}
  188. ( 1 + z ) α = n = 0 ( α n ) z n = 1 + ( α 1 ) z + ( α 2 ) z 2 + . (1+z)^{\alpha}=\sum_{n=0}^{\infty}{\alpha\choose n}z^{n}=1+{\alpha\choose 1}z+% {\alpha\choose 2}z^{2}+\cdots.
  189. 1 ( 1 - z ) α + 1 = n = 0 ( n + α n ) z n \frac{1}{(1-z)^{\alpha+1}}=\sum_{n=0}^{\infty}{n+\alpha\choose n}z^{n}
  190. ( n k ) = ( - 1 ) k ( k - n - 1 k ) {n\choose k}=(-1)^{k}{k-n-1\choose k}
  191. ( ( n k ) ) \left(\!\!{\left({{n}\atop{k}}\right)}\!\!\right)
  192. ( f k ) = ( ( r k ) ) = ( r + k - 1 k ) . {\left({{f}\atop{k}}\right)}=\left(\!\!{\left({{r}\atop{k}}\right)}\!\!\right)% ={\left({{r+k-1}\atop{k}}\right)}.
  193. ( f ) k = f k ¯ = ( f - k + 1 ) ( f - 3 ) ( f - 2 ) ( f - 1 ) f (f)_{k}=f^{\underline{k}}=(f-k+1)\cdots(f-3)\cdot(f-2)\cdot(f-1)\cdot f
  194. \color w h i t e | r ( k ) = r k ¯ = r ( r + 1 ) ( r + 2 ) ( r + 3 ) ( r + k - 1 ) {\color{white}{\big|}}r^{(k)}=\,r^{\overline{k}}=\,r\cdot(r+1)\cdot(r+2)\cdot(% r+3)\cdots(r+k-1)
  195. 17 18 19 20 21 = ( 21 ) 5 = 21 5 ¯ = 17 5 ¯ = 17 ( 5 ) 17\cdot 18\cdot 19\cdot 20\cdot 21=(21)_{5}=21^{\underline{5}}=17^{\overline{5% }}=17^{(5)}
  196. ( f k ) = ( f ) k k ! = ( f - k + 1 ) ( f - 2 ) ( f - 1 ) f 1 2 3 4 5 k {\left({{f}\atop{k}}\right)}=\frac{(f)_{k}}{k!}=\frac{(f-k+1)\cdots(f-2)\cdot(% f-1)\cdot f}{1\cdot 2\cdot 3\cdot 4\cdot 5\cdots k}
  197. ( ( r k ) ) = r ( k ) k ! = r ( r + 1 ) ( r + 2 ) ( r + k - 1 ) 1 2 3 4 5 k \left(\!\!{\left({{r}\atop{k}}\right)}\!\!\right)=\frac{r^{(k)}}{k!}=\frac{r% \cdot(r+1)\cdot(r+2)\cdots(r+k-1)}{1\cdot 2\cdot 3\cdot 4\cdot 5\cdots k}
  198. ( - n k ) = - n - ( n + 1 ) - ( n + k - 2 ) - ( n + k - 1 ) k ! = ( - 1 ) k n ( n + 1 ) ( n + 2 ) ( n + k - 1 ) k ! = ( - 1 ) k ( n + k - 1 k ) = ( - 1 ) k ( ( n k ) ) . \begin{aligned}\displaystyle{\left({{-n}\atop{k}}\right)}&\displaystyle=\frac{% -n\cdot-(n+1)\dots-(n+k-2)\cdot-(n+k-1)}{k!}\\ &\displaystyle=(-1)^{k}\;\frac{n\cdot(n+1)\cdot(n+2)\cdots(n+k-1)}{k!}\\ &\displaystyle=(-1)^{k}{\left({{n+k-1}\atop{k}}\right)}\\ &\displaystyle=(-1)^{k}\left(\!\!{\left({{n}\atop{k}}\right)}\!\!\right)\;.% \end{aligned}
  199. n = - 1 n=-1
  200. ( - 1 ) k = ( - 1 k ) = ( ( - k k ) ) . (-1)^{k}={\left({{-1}\atop{k}}\right)}=\left(\!\!{\left({{-k}\atop{k}}\right)}% \!\!\right)\,.
  201. ( - 4 7 ) \displaystyle{\left({{-4}\atop{7}}\right)}
  202. ( x y ) = Γ ( x + 1 ) Γ ( y + 1 ) Γ ( x - y + 1 ) = 1 ( x + 1 ) B ( x - y + 1 , y + 1 ) . {x\choose y}=\frac{\Gamma(x+1)}{\Gamma(y+1)\Gamma(x-y+1)}=\frac{1}{(x+1)B(x-y+% 1,y+1)}.
  203. Γ \Gamma
  204. ( x y ) = sin ( y π ) sin ( x π ) ( - y - 1 - x - 1 ) = sin ( ( x - y ) π ) sin ( x π ) ( y - x - 1 y ) ; {x\choose y}=\frac{\sin(y\pi)}{\sin(x\pi)}{-y-1\choose-x-1}=\frac{\sin((x-y)% \pi)}{\sin(x\pi)}{y-x-1\choose y};
  205. ( x y ) ( y x ) = sin ( ( x - y ) π ) ( x - y ) π . {x\choose y}\cdot{y\choose x}=\frac{\sin((x-y)\pi)}{(x-y)\pi}.
  206. ( n m ) = ( n n - m ) \textstyle{{n\choose m}={n\choose n-m}}
  207. ( - n m ) = ( - n - n - m ) \textstyle{{-n\choose m}={-n\choose-n-m}}
  208. - n -n
  209. y = x y=x
  210. 0 y x 0\leq y\leq x
  211. 0 x y 0\leq x\leq y
  212. x 0 , y 0 x\geq 0,y\leq 0
  213. x 0 , y 0 x\leq 0,y\geq 0
  214. ( - n , m + 1 ) , ( - n , m ) , ( - n - 1 , m - 1 ) , ( - n - 1 , m ) (-n,m+1),(-n,m),(-n-1,m-1),(-n-1,m)
  215. 0 > x > y 0>x>y
  216. - 1 > y > x + 1 -1>y>x+1
  217. ( α β ) = | { B A : | B | = β } | {\alpha\choose\beta}=|\{B\subseteq A:|B|=\beta\}|
  218. α \alpha
  219. α \alpha
  220. ( α β ) {\alpha\choose\beta}
  221. ( α α ) = 2 α {\alpha\choose\alpha}=2^{\alpha}
  222. α \alpha
  223. ( n k ) {n\choose k}
  224. ( n k ) = 0 {\textstyle\left({{n}\atop{k}}\right)}=0
  225. k < 0 k<0
  226. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}
  227. k < 0 k<0

Binomial_distribution.html

  1. 1 - 6 p ( 1 - p ) n p ( 1 - p ) \frac{1-6p(1-p)}{np(1-p)}
  2. 1 2 log 2 ( 2 π e n p ( 1 - p ) ) + O ( 1 n ) \frac{1}{2}\log_{2}\big(2\pi e\,np(1-p)\big)+O\left(\frac{1}{n}\right)
  3. e e
  4. ( 1 - p + p e t ) n (1-p+pe^{t})^{n}\!
  5. ( 1 - p + p e i t ) n (1-p+pe^{it})^{n}\!
  6. G ( z ) = [ ( 1 - p ) + p z ] n . G(z)=\left[(1-p)+pz\right]^{n}.
  7. g n ( p ) = n p ( 1 - p ) g_{n}(p)=\frac{n}{p(1-p)}
  8. n n
  9. f ( k ; n , p ) = Pr ( X = k ) = ( n k ) p k ( 1 - p ) n - k f(k;n,p)=\Pr(X=k)={n\choose k}p^{k}(1-p)^{n-k}
  10. ( n k ) = n ! k ! ( n - k ) ! {n\choose k}=\frac{n!}{k!(n-k)!}
  11. ( n k ) {n\choose k}
  12. f ( k , n , p ) = f ( n - k , n , 1 - p ) . f(k,n,p)=f(n-k,n,1-p).
  13. f ( k + 1 , n , p ) f ( k , n , p ) = ( n - k ) p ( k + 1 ) ( 1 - p ) \frac{f(k+1,n,p)}{f(k,n,p)}=\frac{(n-k)p}{(k+1)(1-p)}
  14. ( n + 1 ) p - 1 M < ( n + 1 ) p . (n+1)p-1\leq M<(n+1)p.
  15. { p ( n - k ) Prob ( k ) + ( k + 1 ) ( p - 1 ) Prob ( k + 1 ) = 0 , Prob ( 0 ) = ( 1 - p ) n } \left\{p(n-k)\,\text{Prob}(k)+(k+1)(p-1)\,\text{Prob}(k+1)=0,\,\text{Prob}(0)=% (1-p)^{n}\right\}
  16. F ( k ; n , p ) = Pr ( X k ) = i = 0 k ( n i ) p i ( 1 - p ) n - i F(k;n,p)=\Pr(X\leq k)=\sum_{i=0}^{\lfloor k\rfloor}{n\choose i}p^{i}(1-p)^{n-i}
  17. k \scriptstyle\lfloor k\rfloor\,
  18. F ( k ; n , p ) = Pr ( X k ) = I 1 - p ( n - k , k + 1 ) = ( n - k ) ( n k ) 0 1 - p t n - k - 1 ( 1 - t ) k d t . \begin{aligned}\displaystyle F(k;n,p)&\displaystyle=\Pr(X\leq k)\\ &\displaystyle=I_{1-p}(n-k,k+1)\\ &\displaystyle=(n-k){n\choose k}\int_{0}^{1-p}t^{n-k-1}(1-t)^{k}\,dt.\end{aligned}
  19. Pr ( 0 heads ) = f ( 0 ) = Pr ( X = 0 ) = ( 6 0 ) 0.3 0 ( 1 - 0.3 ) 6 - 0 0.1176 \Pr(0\,\text{ heads})=f(0)=\Pr(X=0)={6\choose 0}0.3^{0}(1-0.3)^{6-0}\approx 0.% 1176
  20. Pr ( 1 heads ) = f ( 1 ) = Pr ( X = 1 ) = ( 6 1 ) 0.3 1 ( 1 - 0.3 ) 6 - 1 0.3025 \Pr(1\,\text{ heads})=f(1)=\Pr(X=1)={6\choose 1}0.3^{1}(1-0.3)^{6-1}\approx 0.% 3025
  21. Pr ( 2 heads ) = f ( 2 ) = Pr ( X = 2 ) = ( 6 2 ) 0.3 2 ( 1 - 0.3 ) 6 - 2 0.3241 \Pr(2\,\text{ heads})=f(2)=\Pr(X=2)={6\choose 2}0.3^{2}(1-0.3)^{6-2}\approx 0.% 3241
  22. Pr ( 3 heads ) = f ( 3 ) = Pr ( X = 3 ) = ( 6 3 ) 0.3 3 ( 1 - 0.3 ) 6 - 3 0.1852 \Pr(3\,\text{ heads})=f(3)=\Pr(X=3)={6\choose 3}0.3^{3}(1-0.3)^{6-3}\approx 0.% 1852
  23. Pr ( 4 heads ) = f ( 4 ) = Pr ( X = 4 ) = ( 6 4 ) 0.3 4 ( 1 - 0.3 ) 6 - 4 0.0595 \Pr(4\,\text{ heads})=f(4)=\Pr(X=4)={6\choose 4}0.3^{4}(1-0.3)^{6-4}\approx 0.% 0595
  24. Pr ( 5 heads ) = f ( 5 ) = Pr ( X = 5 ) = ( 6 5 ) 0.3 5 ( 1 - 0.3 ) 6 - 5 0.0102 \Pr(5\,\text{ heads})=f(5)=\Pr(X=5)={6\choose 5}0.3^{5}(1-0.3)^{6-5}\approx 0.% 0102
  25. Pr ( 6 heads ) = f ( 6 ) = Pr ( X = 6 ) = ( 6 6 ) 0.3 6 ( 1 - 0.3 ) 6 - 6 0.0007 \Pr(6\,\text{ heads})=f(6)=\Pr(X=6)={6\choose 6}0.3^{6}(1-0.3)^{6-6}\approx 0.% 0007
  26. E [ X ] = n p , \operatorname{E}[X]=np,
  27. Var [ X ] = n p ( 1 - p ) . \operatorname{Var}[X]=np(1-p).
  28. ( n + 1 ) p \lfloor(n+1)p\rfloor
  29. \lfloor\cdot\rfloor
  30. mode = { ( n + 1 ) p if ( n + 1 ) p is 0 or a noninteger , ( n + 1 ) p and ( n + 1 ) p - 1 if ( n + 1 ) p { 1 , , n } , n if ( n + 1 ) p = n + 1. \,\text{mode}=\begin{cases}\lfloor(n+1)\,p\rfloor&\,\text{if }(n+1)p\,\text{ % is 0 or a noninteger},\\ (n+1)\,p\ \,\text{ and }\ (n+1)\,p-1&\,\text{if }(n+1)p\in\{1,\dots,n\},\\ n&\,\text{if }(n+1)p=n+1.\end{cases}
  31. Cov ( X , Y ) = E ( X Y ) - μ X μ Y . \operatorname{Cov}(X,Y)=\operatorname{E}(XY)-\mu_{X}\mu_{Y}.
  32. Cov ( X , Y ) = p B - p X p Y , \operatorname{Cov}(X,Y)=p_{B}-p_{X}p_{Y},
  33. Cov ( X , Y ) n = n ( p B - p X p Y ) . \operatorname{Cov}(X,Y)_{n}=n(p_{B}-p_{X}p_{Y}).
  34. X + Y B ( n + m , p ) . X+Y\sim B(n+m,p).\,
  35. B ( n + m , p ¯ ) . B(n+m,\bar{p}).\,
  36. Y B ( n , p q ) . Y\sim B(n,pq).
  37. 𝒩 ( n p , n p ( 1 - p ) ) , \mathcal{N}(np,\,np(1-p)),
  38. | ( 1 n ) ( 1 - p p - p 1 - p ) | < 0.3 \left|\left(\frac{1}{\sqrt{n}}\right)\left(\sqrt{\frac{1-p}{p}}-\sqrt{\frac{p}% {1-p}}\right)\right|<0.3
  39. μ ± 3 σ = n p ± 3 n p ( 1 - p ) [ 0 , n ] . \mu\pm 3\sigma=np\pm 3\sqrt{np(1-p)}\in[0,n].
  40. X - n p n p ( 1 - p ) \frac{X-np}{\sqrt{np(1-p)}}
  41. P ( p ; α , β ) = p α - 1 ( 1 - p ) β - 1 B ( α , β ) P(p;\alpha,\beta)=\frac{p^{\alpha-1}(1-p)^{\beta-1}}{\mathrm{B}(\alpha,\beta)}
  42. p ^ = n 1 n \hat{p}=\frac{n_{1}}{n}
  43. p ^ ± z α 2 p ^ ( 1 - p ^ ) n . \hat{p}\pm z_{\frac{\alpha}{2}}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}.
  44. p ~ ± z α 2 p ~ ( 1 - p ~ ) n + z α 2 2 . \tilde{p}\pm z_{\frac{\alpha}{2}}\sqrt{\frac{\tilde{p}(1-\tilde{p})}{n+z_{% \frac{\alpha}{2}}^{2}}}.
  45. p ~ = n 1 + 1 2 z α 2 2 n + z α 2 2 \tilde{p}=\frac{n_{1}+\frac{1}{2}z_{\frac{\alpha}{2}}^{2}}{n+z_{\frac{\alpha}{% 2}}^{2}}
  46. sin 2 ( arcsin ( p ^ ) ± z 2 n ) \sin^{2}\left(\arcsin\left(\sqrt{\hat{p}}\right)\pm\frac{z}{2\sqrt{n}}\right)
  47. p ^ + 1 2 n z 1 - α 2 2 ± 1 2 n z 1 - α 2 4 n p ^ ( 1 - p ^ ) + z 1 - α 2 2 1 + 1 n z 1 - α 2 2 . \frac{\hat{p}+\frac{1}{2n}z_{1-\frac{\alpha}{2}}^{2}\pm\frac{1}{2n}z_{1-\frac{% \alpha}{2}}\sqrt{4n\hat{p}(1-\hat{p})+z_{1-\frac{\alpha}{2}}^{2}}}{1+\frac{1}{% n}z_{1-\frac{\alpha}{2}}^{2}}.
  48. F ( k ; n , p ) exp ( - 2 ( n p - k ) 2 n ) , F(k;n,p)\leq\exp\left(-2\frac{(np-k)^{2}}{n}\right),\!
  49. F ( k ; n , p ) exp ( - 1 2 p ( n p - k ) 2 n ) . F(k;n,p)\leq\exp\left(-\frac{1}{2\,p}\frac{(np-k)^{2}}{n}\right).\!
  50. F ( k ; n , 1 2 ) 1 15 exp ( - 16 ( n 2 - k ) 2 n ) . F(k;n,\tfrac{1}{2})\geq\frac{1}{15}\exp\left(-\frac{16(\frac{n}{2}-k)^{2}}{n}% \right).\!
  51. \rightarrow
  52. F ( k ; n , p ) exp ( - n D ( k n | | p ) ) if 0 < k n < p F(k;n,p)\leq\exp\left(-nD\left(\frac{k}{n}\left|\right|p\right)\right)\quad% \quad\mbox{if }~{}0<\frac{k}{n}<p\!
  53. D ( a | | p ) = ( a ) log a p + ( 1 - a ) log 1 - a 1 - p . D(a||p)=(a)\log\frac{a}{p}+(1-a)\log\frac{1-a}{1-p}.\!
  54. Pr ( X k ) = F ( n - k ; n , 1 - p ) exp ( - n D ( k n | | p ) ) if p < k n < 1. \Pr(X\geq k)=F(n-k;n,1-p)\leq\exp\left(-nD\left(\frac{k}{n}\left|\right|p% \right)\right)\quad\quad\mbox{if }~{}p<\frac{k}{n}<1.\!
  55. Pr ( X k ) = F ( n - k ; n , 1 - p ) 1 ( n + 1 ) 2 exp ( - n D ( k n | | p ) ) if p < k n < 1. \Pr(X\geq k)=F(n-k;n,1-p)\geq\frac{1}{(n+1)^{2}}\exp\left(-nD\left(\frac{k}{n}% \left|\right|p\right)\right)\quad\quad\mbox{if }~{}p<\frac{k}{n}<1.\!

Binomial_theorem.html

  1. b b
  2. c c
  3. b + c = n b+c=n
  4. a a
  5. n n
  6. b b
  7. ( x + y ) 4 = x 4 + 4 x 3 y + 6 x 2 y 2 + 4 x y 3 + y 4 . (x+y)^{4}\;=\;x^{4}\,+\,4x^{3}y\,+\,6x^{2}y^{2}\,+\,4xy^{3}\,+\,y^{4}.
  8. a a
  9. ( n b ) {\textstyle\left({{n}\atop{b}}\right)}
  10. ( n c ) {\textstyle\left({{n}\atop{c}}\right)}
  11. n n
  12. b b
  13. ( n b ) {\textstyle\left({{n}\atop{b}}\right)}
  14. b b
  15. n n
  16. n ! ( n - k ) ! k ! \frac{n!}{(n-k)!k!}
  17. ( 1 + a ) n (1+a)^{n}
  18. ( 1 + a ) n - 1 (1+a)^{n-1}
  19. ( x + y ) n = ( n 0 ) x n y 0 + ( n 1 ) x n - 1 y 1 + ( n 2 ) x n - 2 y 2 + + ( n n - 1 ) x 1 y n - 1 + ( n n ) x 0 y n , (x+y)^{n}={n\choose 0}x^{n}y^{0}+{n\choose 1}x^{n-1}y^{1}+{n\choose 2}x^{n-2}y% ^{2}+\cdots+{n\choose n-1}x^{1}y^{n-1}+{n\choose n}x^{0}y^{n},
  20. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}
  21. ( n 0 ) x n + {\left({{n}\atop{0}}\right)}x^{n}+\ldots
  22. ( x + y ) n = k = 0 n ( n k ) x n - k y k = k = 0 n ( n k ) x k y n - k . (x+y)^{n}=\sum_{k=0}^{n}{n\choose k}x^{n-k}y^{k}=\sum_{k=0}^{n}{n\choose k}x^{% k}y^{n-k}.
  23. ( 1 + x ) n = ( n 0 ) x 0 + ( n 1 ) x 1 + ( n 2 ) x 2 + + ( n n - 1 ) x n - 1 + ( n n ) x n , (1+x)^{n}={n\choose 0}x^{0}+{n\choose 1}x^{1}+{n\choose 2}x^{2}+\cdots+{n% \choose{n-1}}x^{n-1}+{n\choose n}x^{n},
  24. ( 1 + x ) n = k = 0 n ( n k ) x k . (1+x)^{n}=\sum_{k=0}^{n}{n\choose k}x^{k}.
  25. ( x + y ) 2 = x 2 + 2 x y + y 2 . (x+y)^{2}=x^{2}+2xy+y^{2}.\!
  26. ( x + y ) 3 \displaystyle(x+y)^{3}
  27. x 0 = 1 x^{0}=1
  28. ( x + y ) n (x+y)^{n}
  29. y 0 = 1 y^{0}=1
  30. ( x + y ) n (x+y)^{n}
  31. 2 n 2^{n}
  32. n + 1 n+1
  33. ( x + 2 ) 3 = x 3 + 3 x 2 ( 2 ) + 3 x ( 2 ) 2 + 2 3 = x 3 + 6 x 2 + 12 x + 8. \begin{aligned}\displaystyle(x+2)^{3}&\displaystyle=x^{3}+3x^{2}(2)+3x(2)^{2}+% 2^{3}\\ &\displaystyle=x^{3}+6x^{2}+12x+8.\end{aligned}
  34. ( x - y ) 3 = x 3 - 3 x 2 y + 3 x y 2 - y 3 . (x-y)^{3}=x^{3}-3x^{2}y+3xy^{2}-y^{3}.\!
  35. 1 + x = 1 + 1 2 x - 1 8 x 2 + 1 16 x 3 - 5 128 x 4 + 7 256 x 5 - \sqrt{1+x}=\textstyle 1+\frac{1}{2}x-\frac{1}{8}x^{2}+\frac{1}{16}x^{3}-\frac{% 5}{128}x^{4}+\frac{7}{256}x^{5}-\cdots
  36. 1 1 + x = 1 - 1 2 x + 3 8 x 2 - 5 16 x 3 + 35 128 x 4 - 63 256 x 5 + \frac{1}{\sqrt{1+x}}=\textstyle 1-\frac{1}{2}x+\frac{3}{8}x^{2}-\frac{5}{16}x^% {3}+\frac{35}{128}x^{4}-\frac{63}{256}x^{5}+\cdots
  37. | x | < 1 |x|<1
  38. ( 1 + x ) - 1 = 1 1 + x = 1 - x + x 2 - x 3 + x 4 - x 5 + (1+x)^{-1}=\frac{1}{1+x}=1-x+x^{2}-x^{3}+x^{4}-x^{5}+\cdots
  39. ( x n ) = n x n - 1 : (x^{n})^{\prime}=nx^{n-1}:
  40. a = x a=x
  41. b = Δ x , b=\Delta x,
  42. ( x + Δ x ) n , (x+\Delta x)^{n},
  43. Δ x \Delta x
  44. n x n - 1 , nx^{n-1},
  45. ( n - 1 ) : (n-1):
  46. ( x + Δ x ) n = x n + n x n - 1 Δ x + ( n 2 ) x n - 2 ( Δ x ) 2 + . (x+\Delta x)^{n}=x^{n}+nx^{n-1}\Delta x+{\textstyle\left({{n}\atop{2}}\right)}% x^{n-2}(\Delta x)^{2}+\cdots.
  47. ( Δ x ) 2 (\Delta x)^{2}
  48. ( x n ) = n x n - 1 , (x^{n})^{\prime}=nx^{n-1},
  49. ( n - 1 ) (n-1)
  50. x n - 1 d x = 1 n x n \textstyle{\int x^{n-1}\,dx=\tfrac{1}{n}x^{n}}
  51. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}
  52. ( n k ) = n ! k ! ( n - k ) ! {n\choose k}=\frac{n!}{k!\,(n-k)!}
  53. ( n k ) = n ( n - 1 ) ( n - k + 1 ) k ( k - 1 ) 1 = = 1 k n - + 1 = = 0 k - 1 n - k - {n\choose k}=\frac{n(n-1)\cdots(n-k+1)}{k(k-1)\cdots 1}=\prod_{\ell=1}^{k}% \frac{n-\ell+1}{\ell}=\prod_{\ell=0}^{k-1}\frac{n-\ell}{k-\ell}
  54. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}
  55. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}
  56. ( x + y ) ( x + y ) ( x + y ) ( x + y ) , (x+y)(x+y)(x+y)\cdots(x+y),
  57. ( x + y ) 3 = ( x + y ) ( x + y ) ( x + y ) = x x x + x x y + x y x + x y y ¯ + y x x + y x y ¯ + y y x ¯ + y y y = x 3 + 3 x 2 y + 3 x y 2 ¯ + y 3 . \begin{aligned}\displaystyle(x+y)^{3}&\displaystyle=(x+y)(x+y)(x+y)\\ &\displaystyle=xxx+xxy+xyx+\underline{xyy}+yxx+\underline{yxy}+\underline{yyx}% +yyy\\ &\displaystyle=x^{3}+3x^{2}y+\underline{3xy^{2}}+y^{3}.\end{aligned}\,
  58. ( 3 2 ) = 3 {\textstyle\left({{3}\atop{2}}\right)}=3
  59. x y y , y x y , y y x , xyy,\;yxy,\;yyx,
  60. { 2 , 3 } , { 1 , 3 } , { 1 , 2 } , \{2,3\},\;\{1,3\},\;\{1,2\},
  61. ( n k ) {n\choose k}
  62. ( n k ) {n\choose k}
  63. n ! k ! ( n - k ) ! \frac{n!}{k!\,(n-k)!}
  64. ( 0 0 ) = 1 {\textstyle\left({{0}\atop{0}}\right)}=1
  65. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}
  66. ( x + y ) n + 1 = x ( x + y ) n + y ( x + y ) n , (x+y)^{n+1}=x(x+y)^{n}+y(x+y)^{n},\,
  67. [ ( x + y ) n + 1 ] j , k = [ ( x + y ) n ] j - 1 , k + [ ( x + y ) n ] j , k - 1 , [(x+y)^{n+1}]_{j,k}=[(x+y)^{n}]_{j-1,k}+[(x+y)^{n}]_{j,k-1},
  68. ( n k ) + ( n k - 1 ) = ( n + 1 k ) , {\left({{n}\atop{k}}\right)}+{\left({{n}\atop{k-1}}\right)}={\left({{n+1}\atop% {k}}\right)},
  69. ( x + y ) n + 1 = k = 0 n + 1 ( n + 1 k ) x n + 1 - k y k , (x+y)^{n+1}=\sum_{k=0}^{n+1}{\textstyle\left({{n+1}\atop{k}}\right)}x^{n+1-k}y% ^{k},
  70. ( r k ) = r ( r - 1 ) ( r - k + 1 ) k ! = ( r ) k k ! , {r\choose k}=\frac{r\,(r-1)\cdots(r-k+1)}{k!}=\frac{(r)_{k}}{k!},
  71. ( ) k (\cdot)_{k}
  72. ( x + y ) r = k = 0 ( r k ) x r - k y k ( 2 ) = x r + r x r - 1 y + r ( r - 1 ) 2 ! x r - 2 y 2 + r ( r - 1 ) ( r - 2 ) 3 ! x r - 3 y 3 + . \begin{aligned}\displaystyle(x+y)^{r}&\displaystyle=\sum_{k=0}^{\infty}{r% \choose k}x^{r-k}y^{k}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad(2% )\\ &\displaystyle=x^{r}+rx^{r-1}y+\frac{r(r-1)}{2!}x^{r-2}y^{2}+\frac{r(r-1)(r-2)% }{3!}x^{r-3}y^{3}+\cdots.\end{aligned}
  73. 1 ( 1 - x ) s = k = 0 ( s + k - 1 k ) x k k = 0 ( s + k - 1 s - 1 ) x k . \frac{1}{(1-x)^{s}}=\sum_{k=0}^{\infty}{s+k-1\choose k}x^{k}\equiv\sum_{k=0}^{% \infty}{s+k-1\choose s-1}x^{k}.
  74. ( x 1 + x 2 + + x m ) n = k 1 + k 2 + + k m = n ( n k 1 , k 2 , , k m ) x 1 k 1 x 2 k 2 x m k m . (x_{1}+x_{2}+\cdots+x_{m})^{n}=\sum_{k_{1}+k_{2}+\cdots+k_{m}=n}{n\choose k_{1% },k_{2},\ldots,k_{m}}x_{1}^{k_{1}}x_{2}^{k_{2}}\cdots x_{m}^{k_{m}}.
  75. ( n k 1 , , k m ) {\textstyle\left({{n}\atop{k_{1},\cdots,k_{m}}}\right)}
  76. ( n k 1 , k 2 , , k m ) = n ! k 1 ! k 2 ! k m ! . {n\choose k_{1},k_{2},\ldots,k_{m}}=\frac{n!}{k_{1}!\,k_{2}!\cdots k_{m}!}.
  77. ( n k 1 , , k m ) {\textstyle\left({{n}\atop{k_{1},\cdots,k_{m}}}\right)}
  78. ( x 1 + y 1 ) n 1 ( x d + y d ) n d = k 1 = 0 n 1 k d = 0 n d ( n 1 k 1 ) x 1 k 1 y 1 n 1 - k 1 ( n d k d ) x d k d y d n d - k d . (x_{1}+y_{1})^{n_{1}}\cdots(x_{d}+y_{d})^{n_{d}}=\sum_{k_{1}=0}^{n_{1}}\cdots% \sum_{k_{d}=0}^{n_{d}}{\left({{n_{1}}\atop{k_{1}}}\right)}\,x_{1}^{k_{1}}y_{1}% ^{n_{1}-k_{1}}\;\ldots\;{\left({{n_{d}}\atop{k_{d}}}\right)}\,x_{d}^{k_{d}}y_{% d}^{n_{d}-k_{d}}.
  79. ( x + y ) α = ν α ( α ν ) x ν y α - ν . (x+y)^{\alpha}=\sum_{\nu\leq\alpha}{\left({{\alpha}\atop{\nu}}\right)}\,x^{\nu% }y^{\alpha-\nu}.
  80. cos ( n x ) + i sin ( n x ) = ( cos x + i sin x ) n . \cos\left(nx\right)+i\sin\left(nx\right)=\left(\cos x+i\sin x\right)^{n}.\,
  81. ( cos x + i sin x ) 2 = cos 2 x + 2 i cos x sin x - sin 2 x , \left(\cos x+i\sin x\right)^{2}=\cos^{2}x+2i\cos x\sin x-\sin^{2}x,
  82. cos ( 2 x ) = cos 2 x - sin 2 x and sin ( 2 x ) = 2 cos x sin x , \cos(2x)=\cos^{2}x-\sin^{2}x\quad\,\text{and}\quad\sin(2x)=2\cos x\sin x,
  83. ( cos x + i sin x ) 3 = cos 3 x + 3 i cos 2 x sin x - 3 cos x sin 2 x - i sin 3 x , \left(\cos x+i\sin x\right)^{3}=\cos^{3}x+3i\cos^{2}x\sin x-3\cos x\sin^{2}x-i% \sin^{3}x,
  84. cos ( 3 x ) = cos 3 x - 3 cos x sin 2 x and sin ( 3 x ) = 3 cos 2 x sin x - sin 3 x . \cos(3x)=\cos^{3}x-3\cos x\sin^{2}x\quad\,\text{and}\quad\sin(3x)=3\cos^{2}x% \sin x-\sin^{3}x.
  85. cos ( n x ) = k even ( - 1 ) k / 2 ( n k ) cos n - k x sin k x \cos(nx)=\sum_{k\,\text{ even}}(-1)^{k/2}{n\choose k}\cos^{n-k}x\sin^{k}x
  86. sin ( n x ) = k odd ( - 1 ) ( k - 1 ) / 2 ( n k ) cos n - k x sin k x . \sin(nx)=\sum_{k\,\text{ odd}}(-1)^{(k-1)/2}{n\choose k}\cos^{n-k}x\sin^{k}x.
  87. e = lim n ( 1 + 1 n ) n . e=\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n}.
  88. ( 1 + 1 n ) n = 1 + ( n 1 ) 1 n + ( n 2 ) 1 n 2 + ( n 3 ) 1 n 3 + + ( n n ) 1 n n . \left(1+\frac{1}{n}\right)^{n}=1+{n\choose 1}\frac{1}{n}+{n\choose 2}\frac{1}{% n^{2}}+{n\choose 3}\frac{1}{n^{3}}+\cdots+{n\choose n}\frac{1}{n^{n}}.
  89. ( n k ) 1 n k = 1 k ! n ( n - 1 ) ( n - 2 ) ( n - k + 1 ) n k {n\choose k}\frac{1}{n^{k}}\;=\;\frac{1}{k!}\cdot\frac{n(n-1)(n-2)\cdots(n-k+1% )}{n^{k}}
  90. lim n ( n k ) 1 n k = 1 k ! . \lim_{n\to\infty}{n\choose k}\frac{1}{n^{k}}=\frac{1}{k!}.
  91. e = 1 0 ! + 1 1 ! + 1 2 ! + 1 3 ! + . e=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\cdots.

Bioleaching.html

  1. FeS 2 + 6 Fe 3 + + 3 H 2 O 7 Fe 2 + + S 2 O 3 2 - + 6 H + \mathrm{FeS_{2}+6\ Fe^{\,3+}+3\ H_{2}O\longrightarrow 7\ Fe^{\,2+}+S_{2}O_{3}^% {\,2-}+6\ H^{+}}
  2. 4 Fe 2 + + O 2 + 4 H + 4 Fe 3 + + 2 H 2 O \mathrm{4\ Fe^{\,2+}+\ O_{2}+4\ H^{+}\longrightarrow 4\ Fe^{\,3+}+2\ H_{2}O}
  3. S 2 O 3 2 - + 2 O 2 + H 2 O 2 SO 4 2 - + 2 H + \mathrm{S_{2}O_{3}^{\,2-}+2\ O_{2}+H_{2}O\longrightarrow 2\ SO_{4}^{\,2-}+2\ H% ^{+}}
  4. 2 FeS 2 + 7 O 2 + 2 H 2 O 2 Fe 2 + + 4 SO 4 2 - + 4 H + \mathrm{2\ FeS_{2}+7\ O_{2}+2\ H_{2}O\longrightarrow 2\ Fe^{\,2+}+4\ SO_{4}^{% \,2-}+4\ H^{+}}
  5. CuFeS 2 + 4 Fe 3 + Cu 2 + + 5 Fe 2 + + 2 S 0 \mathrm{CuFeS_{2}+4\ Fe^{\,3+}\longrightarrow Cu^{\,2+}+5\ Fe^{\,2+}+2\ S_{0}}
  6. 4 Fe 2 + + O 2 + 4 H + 4 Fe 3 + + 2 H 2 O \mathrm{4\ Fe^{\,2+}+O_{2}+4\ H^{+}\longrightarrow 4\ Fe^{\,3+}+2\ H_{2}O}
  7. 2 S 0 + 3 O 2 + 2 H 2 O 2 SO 4 2 - + 4 H + \mathrm{2\ S^{0}+3\ O_{2}+2\ H_{2}O\longrightarrow 2\ SO_{4}^{\,2-}+4\ H^{+}}
  8. CuFeS 2 + 4 O 2 Cu 2 + + Fe 2 + + 2 SO 4 2 - \mathrm{CuFeS_{2}+4\ O_{2}\longrightarrow Cu^{\,2+}+Fe^{\,2+}+2\ SO_{4}^{\,2-}}

Biorhythm.html

  1. sin ( 2 π t / 23 ) \sin(2\pi t/23)
  2. sin ( 2 π t / 28 ) \sin(2\pi t/28)
  3. sin ( 2 π t / 33 ) \sin(2\pi t/33)
  4. t t

Bipolar_junction_transistor.html

  1. β F \beta_{F}
  2. α F = I C I E \alpha_{F}=\frac{I_{\,\text{C}}}{I_{\,\text{E}}}
  3. β F = I C I B \beta_{F}=\frac{I_{\,\text{C}}}{I_{\,\text{B}}}
  4. β F = α F 1 - α F α F = β F β F + 1 \beta_{F}=\frac{\alpha_{F}}{1-\alpha_{F}}\iff\alpha_{F}=\frac{\beta_{F}}{\beta% _{F}+1}
  5. I C I_{C}
  6. U C E U_{CE}
  7. I B I_{B}
  8. β F \beta_{F}
  9. h FE h_{\,\text{FE}}
  10. h fe h_{\,\text{fe}}
  11. h fe h_{\,\text{fe}}
  12. β \beta
  13. h FE h_{\,\text{FE}}
  14. h fe h_{\,\text{fe}}
  15. V BE V_{\,\text{BE}}
  16. V BE V_{\,\text{BE}}
  17. V EB V_{\,\text{EB}}
  18. V CE V_{\,\text{CE}}
  19. I C I_{\,\text{C}}
  20. I B I_{\,\text{B}}
  21. I E I_{\,\text{E}}
  22. h FE h_{\,\text{FE}}
  23. h fe h_{\,\text{fe}}
  24. β \beta
  25. V EB V_{\,\text{EB}}
  26. V EB V_{\,\text{EB}}
  27. V BE V_{\,\text{BE}}
  28. V CB V_{\,\text{CB}}
  29. I E = I ES ( e V BE V T - 1 ) I_{\,\text{E}}=I_{\,\text{ES}}\left(e^{\frac{V_{\,\text{BE}}}{V_{\,\text{T}}}}% -1\right)
  30. I C = α F I E I_{\,\text{C}}=\alpha_{F}I_{\,\text{E}}
  31. I B = ( 1 - α F ) I E I_{\,\text{B}}=\left(1-\alpha_{F}\right)I_{\,\text{E}}
  32. J n ( base ) = q D n n b o W e V EB V T J_{n\,(\,\text{base})}=\frac{qD_{n}n_{bo}}{W}e^{\frac{V_{\,\text{EB}}}{V_{\,% \text{T}}}}
  33. V T V_{\,\text{T}}
  34. k T / q kT/q
  35. I E I_{\,\text{E}}
  36. I C I_{\,\text{C}}
  37. α F \alpha_{F}
  38. I ES I_{\,\text{ES}}
  39. V BE V_{\,\text{BE}}
  40. D n D_{n}
  41. α \alpha
  42. β \beta
  43. β \beta
  44. i C = I S ( e V BE V T - e V BC V T ) - I S β R ( e V BC V T - 1 ) i_{\,\text{C}}=I_{\,\text{S}}\left(e^{\frac{V_{\,\text{BE}}}{V_{\,\text{T}}}}-% e^{\frac{V_{\,\text{BC}}}{V_{\,\text{T}}}}\right)-\frac{I_{\,\text{S}}}{\beta_% {R}}\left(e^{\frac{V_{\,\text{BC}}}{V_{\,\text{T}}}}-1\right)
  45. i B = I S β F ( e V BE V T - 1 ) + I S β R ( e V BC V T - 1 ) i_{\,\text{B}}=\frac{I_{\,\text{S}}}{\beta_{F}}\left(e^{\frac{V_{\,\text{BE}}}% {V_{\,\text{T}}}}-1\right)+\frac{I_{\,\text{S}}}{\beta_{R}}\left(e^{\frac{V_{% \,\text{BC}}}{V_{\,\text{T}}}}-1\right)
  46. i E = I S ( e V BE V T - e V BC V T ) + I S β F ( e V BE V T - 1 ) i_{\,\text{E}}=I_{\,\text{S}}\left(e^{\frac{V_{\,\text{BE}}}{V_{\,\text{T}}}}-% e^{\frac{V_{\,\text{BC}}}{V_{\,\text{T}}}}\right)+\frac{I_{\,\text{S}}}{\beta_% {F}}\left(e^{\frac{V_{\,\text{BE}}}{V_{\,\text{T}}}}-1\right)
  47. i C i_{\,\text{C}}
  48. i B i_{\,\text{B}}
  49. i E i_{\,\text{E}}
  50. β F \beta_{F}
  51. β R \beta_{R}
  52. I S I_{\,\text{S}}
  53. V T V_{\,\text{T}}
  54. V BE V_{\,\text{BE}}
  55. V BC V_{\,\text{BC}}
  56. V CB = V CE - V BE V_{\,\text{CB}}=V_{\,\text{CE}}-V_{\,\text{BE}}
  57. i C i_{\,\text{C}}
  58. β F \beta_{F}
  59. i C = I S e v BE V T ( 1 + V CE V A ) i_{\,\text{C}}=I_{\,\text{S}}\,e^{\frac{v_{\,\text{BE}}}{V_{\,\text{T}}}}\left% (1+\frac{V_{\,\text{CE}}}{V_{\,\text{A}}}\right)
  60. β F = β F 0 ( 1 + V CB V A ) \beta_{F}=\beta_{F0}\left(1+\frac{V_{\,\text{CB}}}{V_{\,\text{A}}}\right)
  61. r o = V A I C r_{\,\text{o}}=\frac{V_{\,\text{A}}}{I_{\,\text{C}}}
  62. V CE V_{\,\text{CE}}
  63. V A V_{\,\text{A}}
  64. β F 0 \beta_{F0}
  65. V CB V_{\,\text{CB}}
  66. r o r_{\,\text{o}}
  67. I C I_{\,\text{C}}
  68. β \beta

Bipyramid.html

  1. V = 2 3 B h \scriptstyle{V=}\tfrac{2}{3}\scriptstyle{Bh}
  2. V = n 6 h s 2 cot π n . V=\frac{n}{6}hs^{2}\cot\frac{\pi}{n}.
  3. 2 3 \scriptstyle\frac{2}{3}
  4. - 1 7 \scriptstyle-\frac{1}{7}
  5. - 1 7 \scriptstyle-\frac{1}{7}
  6. 2 3 \scriptstyle\frac{\sqrt{2}}{3}
  7. - 2 5 \scriptstyle-\frac{2}{5}
  8. 1 5 \scriptstyle\frac{1}{5}
  9. 2 2 3 \scriptstyle\frac{2\sqrt{2}}{3}
  10. 1 11 \scriptstyle\frac{1}{11}
  11. - 5 11 \scriptstyle-\frac{5}{11}
  12. 5 - 1 3 \scriptstyle\frac{\sqrt{5}-1}{3}
  13. - 10 + 9 5 61 \scriptstyle-\frac{10+9\sqrt{5}}{61}
  14. 12 5 - 7 61 \scriptstyle\frac{12\sqrt{5}-7}{61}
  15. 2 \scriptstyle\sqrt{2}
  16. - 1 3 \scriptstyle-\frac{1}{3}
  17. - 1 3 \scriptstyle-\frac{1}{3}
  18. - 1 2 \scriptstyle-\frac{1}{2}
  19. 5 + 3 5 5 \scriptstyle\frac{5+3\sqrt{5}}{5}
  20. - 11 + 4 5 41 \scriptstyle-\frac{11+4\sqrt{5}}{41}
  21. - 11 + 4 5 41 \scriptstyle-\frac{11+4\sqrt{5}}{41}

Bistability.html

  1. d y d t = y ( 1 - y 2 ) . \frac{dy}{dt}=y(1-y^{2}).
  2. y 4 4 - y 2 2 \frac{y^{4}}{4}-\frac{y^{2}}{2}
  3. y = 1 y=1
  4. y = 0 y=0
  5. y = - 1 y=-1
  6. y = 0 y=0
  7. y ( t ) y(t)
  8. y ( 0 ) y(0)
  9. y ( 0 ) > 0 y(0)>0
  10. y ( t ) y(t)
  11. y ( 0 ) < 0 y(0)<0
  12. y ( t ) y(t)

Bit_error_rate.html

  1. p p = 1 - ( 1 - p e ) N p_{p}=1-(1-p_{e})^{N}
  2. p p p e N . p_{p}\approx p_{e}N.
  3. BER = 1 2 erfc ( E b / N 0 ) \operatorname{BER}=\frac{1}{2}\operatorname{erfc}(\sqrt{E_{b}/N_{0}})
  4. w ( t ) w(t)
  5. x 1 ( t ) = A + w ( t ) x_{1}(t)=A+w(t)
  6. x 0 ( t ) = - A + w ( t ) x_{0}(t)=-A+w(t)
  7. x 1 ( t ) x_{1}(t)
  8. x 0 ( t ) x_{0}(t)
  9. T T
  10. N 0 2 \frac{N_{0}}{2}
  11. x 1 ( t ) x_{1}(t)
  12. 𝒩 ( A , N 0 2 T ) \mathcal{N}\left(A,\frac{N_{0}}{2T}\right)
  13. x 0 ( t ) x_{0}(t)
  14. 𝒩 ( - A , N 0 2 T ) \mathcal{N}\left(-A,\frac{N_{0}}{2T}\right)
  15. p e = p ( 0 | 1 ) p 1 + p ( 1 | 0 ) p 0 p_{e}=p(0|1)p_{1}+p(1|0)p_{0}
  16. p ( 1 | 0 ) = 0.5 erfc ( A + λ N o / T ) p(1|0)=0.5\,\operatorname{erfc}\left(\frac{A+\lambda}{\sqrt{N_{o}/T}}\right)
  17. p ( 0 | 1 ) = 0.5 erfc ( A - λ N o / T ) p(0|1)=0.5\,\operatorname{erfc}\left(\frac{A-\lambda}{\sqrt{N_{o}/T}}\right)
  18. λ \lambda
  19. p 1 = p 0 = 0.5 p_{1}=p_{0}=0.5
  20. E = A 2 T E=A^{2}T
  21. p e = 0.5 erfc ( E N o ) . p_{e}=0.5\,\operatorname{erfc}\left(\sqrt{\frac{E}{N_{o}}}\right).

Black_body.html

  1. T = c 3 8 π G k B M , T=\frac{\hbar c^{3}}{8\pi Gk_{B}M}\ ,
  2. P / A = σ T 4 , P/A=\sigma T^{4}\ ,

Black_hole.html

  1. Q 2 + ( J M ) 2 M 2 Q^{2}+\left(\tfrac{J}{M}\right)^{2}\leq M^{2}\,
  2. r sh = 2 G M c 2 2.95 M M Sun km , r_{\mathrm{sh}}=\frac{2GM}{c^{2}}\approx 2.95\,\frac{M}{M_{\mathrm{Sun}}}~{}% \mathrm{km,}
  3. ħ c / G \sqrt{ħc/G}

Blackboard_bold.html

  1. 𝔸 \mathbb{A}
  2. 𝔹 \mathbb{B}
  3. \mathbb{C}
  4. 𝔻 \mathbb{D}
  5. D D D\!\!\!\!D
  6. d d d\!\!\!\!d
  7. 𝔼 \mathbb{E}
  8. e e e\!\!e
  9. 𝔽 \mathbb{F}
  10. 𝔾 \mathbb{G}
  11. \mathbb{H}
  12. 𝕀 \mathbb{I}
  13. i i i\!i
  14. 𝕁 \mathbb{J}
  15. j j j\!\!j
  16. 𝕂 \mathbb{K}
  17. 𝕃 \mathbb{L}
  18. 𝕄 \mathbb{M}
  19. \mathbb{N}
  20. 𝕆 \mathbb{O}
  21. \mathbb{P}
  22. \mathbb{Q}
  23. \mathbb{R}
  24. 𝕊 \mathbb{S}
  25. 𝕋 \mathbb{T}
  26. 𝕌 \mathbb{U}
  27. 𝕍 \mathbb{V}
  28. 𝕎 \mathbb{W}
  29. 𝕏 \mathbb{X}
  30. 𝕐 \mathbb{Y}
  31. \mathbb{Z}

Blaise_Pascal.html

  1. t m n = ( m + n ) ( m + n - 1 ) ( m + 1 ) n ( n - 1 ) 1 . t_{mn}=\frac{(m+n)(m+n-1)\cdots(m+1)}{n(n-1)\cdots 1}.

Block_cipher.html

  1. E K ( P ) := E ( K , P ) : { 0 , 1 } k × { 0 , 1 } n { 0 , 1 } n , E_{K}(P):=E(K,P):\{0,1\}^{k}\times\{0,1\}^{n}\rightarrow\{0,1\}^{n},
  2. E K - 1 ( C ) := D K ( C ) = D ( K , C ) : { 0 , 1 } k × { 0 , 1 } n { 0 , 1 } n , E_{K}^{-1}(C):=D_{K}(C)=D(K,C):\{0,1\}^{k}\times\{0,1\}^{n}\rightarrow\{0,1\}^% {n},
  3. K : D K ( E K ( P ) ) = P . \forall K:D_{K}(E_{K}(P))=P.
  4. ( 2 n ) ! (2^{n})!
  5. M i = R K i ( M i - 1 ) M_{i}=R_{K_{i}}(M_{i-1})
  6. M 0 M_{0}
  7. M r M_{r}
  8. M 0 = M K 0 M_{0}=M\oplus K_{0}
  9. M i = R K i ( M i - 1 ) ; i = 1 r M_{i}=R_{K_{i}}(M_{i-1})\;;\;i=1\dots r
  10. C = M r K r + 1 C=M_{r}\oplus K_{r+1}
  11. F {\rm F}
  12. K 0 , K 1 , , K n K_{0},K_{1},\ldots,K_{n}
  13. 0 , 1 , , n 0,1,\ldots,n
  14. L 0 L_{0}
  15. R 0 R_{0}
  16. i = 0 , 1 , , n i=0,1,\dots,n
  17. L i + 1 = R i L_{i+1}=R_{i}\,
  18. R i + 1 = L i F ( R i , K i ) R_{i+1}=L_{i}\oplus{\rm F}(R_{i},K_{i})
  19. ( R n + 1 , L n + 1 ) (R_{n+1},L_{n+1})
  20. ( R n + 1 , L n + 1 ) (R_{n+1},L_{n+1})
  21. i = n , n - 1 , , 0 i=n,n-1,\ldots,0
  22. R i = L i + 1 R_{i}=L_{i+1}\,
  23. L i = R i + 1 F ( L i + 1 , K i ) L_{i}=R_{i+1}\oplus{\rm F}(L_{i+1},K_{i})
  24. ( L 0 , R 0 ) (L_{0},R_{0})
  25. F {\rm F}
  26. F \mathrm{F}
  27. F \mathrm{F}
  28. H \mathrm{H}
  29. K 0 , K 1 , , K n K_{0},K_{1},\ldots,K_{n}
  30. 0 , 1 , , n 0,1,\ldots,n
  31. L 0 L_{0}
  32. R 0 R_{0}
  33. i = 0 , 1 , , n i=0,1,\dots,n
  34. ( L i + 1 , R i + 1 ) = H ( L i + T i , R i + T i ) (L_{i+1}^{\prime},R_{i+1}^{\prime})=\mathrm{H}(L_{i}^{\prime}+T_{i},R_{i}^{% \prime}+T_{i})
  35. T i = F ( L i - R i , K i ) T_{i}=\mathrm{F}(L_{i}^{\prime}-R_{i}^{\prime},K_{i})
  36. ( L 0 , R 0 ) = H ( L 0 , R 0 ) (L_{0}^{\prime},R_{0}^{\prime})=\mathrm{H}(L_{0},R_{0})
  37. ( L n + 1 , R n + 1 ) = ( L n + 1 , R n + 1 ) (L_{n+1},R_{n+1})=(L_{n+1}^{\prime},R_{n+1}^{\prime})
  38. ( L n + 1 , R n + 1 ) (L_{n+1},R_{n+1})
  39. i = n , n - 1 , , 0 i=n,n-1,\ldots,0
  40. ( L i , R i ) = H - 1 ( L i + 1 - T i , R i + 1 - T i ) (L_{i}^{\prime},R_{i}^{\prime})=\mathrm{H}^{-1}(L_{i+1}^{\prime}-T_{i},R_{i+1}% ^{\prime}-T_{i})
  41. T i = F ( L i + 1 - R i + 1 , K i ) T_{i}=\mathrm{F}(L_{i+1}^{\prime}-R_{i+1}^{\prime},K_{i})
  42. ( L n + 1 , R n + 1 ) = H - 1 ( L n + 1 , R n + 1 ) (L_{n+1}^{\prime},R_{n+1}^{\prime})=\mathrm{H}^{-1}(L_{n+1},R_{n+1})
  43. ( L 0 , R 0 ) = ( L 0 , R 0 ) (L_{0},R_{0})=(L_{0}^{\prime},R_{0}^{\prime})
  44. π \pi
  45. π \pi

Blood_alcohol_content.html

  1. E B A C = 0.806 S D 1.2 B W W t - ( M R D P ) EBAC=\frac{0.806\cdot SD\cdot 1.2}{BW\cdot Wt}-(MR\cdot DP)
  2. E B A C = ( 0.806 3 1.2 ) / ( 0.58 80 ) - ( 0.015 2 ) = 0.032534483 0.033 g / d L EBAC=(0.806\cdot 3\cdot 1.2)/(0.58\cdot 80)-(0.015\cdot 2)=0.032534483\approx 0% .033g/dL
  3. E B A C = ( 0.806 2.5 1.2 ) / ( 0.49 70 ) - ( 0.017 2 ) = 0.036495627 0.037 g / d L EBAC=(0.806\cdot 2.5\cdot 1.2)/(0.49\cdot 70)-(0.017\cdot 2)=0.036495627% \approx 0.037g/dL

Blood_pressure.html

  1. MAP = ( CO SVR ) + CVP . \!\,\text{MAP}=(\,\text{CO}\cdot\,\text{SVR})+\,\text{CVP}.
  2. P sys \!P_{\,\text{sys}}
  3. P dias \!P_{\,\text{dias}}
  4. MAP P dias + 1 3 ( P sys - P dias ) . \!\,\text{MAP}\approxeq P_{\,\text{dias}}+\frac{1}{3}(P_{\,\text{sys}}-P_{\,% \text{dias}}).
  5. P pulse = P sys - P dias . \!P_{\,\text{pulse}}=P_{\,\text{sys}}-P_{\,\text{dias}}.

Bluff_(poker).html

  1. x = s / ( 1 + s ) x=s/(1+s)
  2. x = 1 / ( 1 + s ) = 50 % x=1/(1+s)=50\%

Blum_Blum_Shub.html

  1. x n + 1 = x n 2 mod M x_{n+1}=x_{n}^{2}\bmod M
  2. x i = ( x 0 2 i mod λ ( M ) ) mod M x_{i}=\left(x_{0}^{2^{i}\bmod\lambda(M)}\right)\bmod M
  3. λ \lambda
  4. λ ( M ) = λ ( p q ) = lcm ( p - 1 , q - 1 ) \lambda(M)=\lambda(p\cdot q)=\operatorname{lcm}(p-1,q-1)
  5. p = 11 p=11
  6. q = 19 q=19
  7. s = 3 s=3
  8. s s
  9. gcd ( φ ( p - 1 ) , φ ( q - 1 ) ) = 2 {\rm gcd}(\varphi(p-1),\varphi(q-1))=2
  10. x 0 x_{0}
  11. x - 1 = s x_{-1}=s
  12. x 0 x_{0}
  13. x 1 x_{1}
  14. x 2 x_{2}
  15. \ldots
  16. x 5 x_{5}

Body_mass_index.html

  1. BMI = masskg heightm 2 = masslb heightin 2 × 703 \mathrm{BMI}=\frac{\,\text{mass}\text{kg}}{\,\text{height}\text{m}^{2}}=\frac{% \,\text{mass}\text{lb}}{\,\text{height}\text{in}^{2}}\times 703

Bohr_model.html

  1. Δ E = E 2 - E 1 = h ν , \Delta{E}=E_{2}-E_{1}=h\nu\ ,
  2. ν = 1 T . \nu={1\over T}.
  3. L = n h 2 π = n L=n{h\over 2\pi}=n\hbar
  4. n λ = 2 π r . n\lambda=2\pi r.\,
  5. m e v 2 r = Z k e e 2 r 2 {m_{\mathrm{e}}v^{2}\over r}={Zk_{\mathrm{e}}e^{2}\over r^{2}}
  6. v = Z k e e 2 m e r . v=\sqrt{Zk_{\mathrm{e}}e^{2}\over m_{\mathrm{e}}r}.
  7. E = 1 2 m e v 2 - Z k e e 2 r = - Z k e e 2 2 r . E={1\over 2}m_{\mathrm{e}}v^{2}-{Zk_{\mathrm{e}}e^{2}\over r}=-{Zk_{\mathrm{e}% }e^{2}\over 2r}.
  8. m e v r = n m_{\mathrm{e}}vr=n\hbar
  9. Δ E n = h T ( E n ) . \Delta E_{n}={h\over T(E_{n})}.
  10. E n E_{n}
  11. E n + 1 E_{n+1}
  12. Δ E 1 r 3 2 E 3 2 . \Delta E\propto{1\over r^{3\over 2}}\propto E^{3\over 2}.
  13. r ¯ \overline{r}
  14. E 1 r 1 L 2 E\propto{1\over r}\propto{1\over L^{2}}
  15. Δ E 1 ( L + ) 2 - 1 L 2 - 2 L 3 - E 3 2 . \Delta E\propto{1\over(L+\hbar)^{2}}-{1\over L^{2}}\approx-{2\hbar\over L^{3}}% \propto-E^{3\over 2}.
  16. L = n h 2 π = n . L={nh\over 2\pi}=n\hbar~{}.
  17. m e k e Z e 2 m e r r = n m_{\,\text{e}}\sqrt{\dfrac{k_{\,\text{e}}Ze^{2}}{m_{\,\text{e}}r}}r=n\hbar
  18. r n = n 2 2 Z k e e 2 m e r_{n}={n^{2}\hbar^{2}\over Zk_{\mathrm{e}}e^{2}m_{\mathrm{e}}}
  19. r 1 = 2 k e e 2 m e 5.29 × 10 - 11 m r_{1}={\hbar^{2}\over k_{\mathrm{e}}e^{2}m_{\mathrm{e}}}\approx 5.29\times 10^% {-11}\mathrm{m}
  20. E = - Z k e e 2 2 r n = - Z 2 ( k e e 2 ) 2 m e 2 2 n 2 - 13.6 Z 2 n 2 eV E=-{Zk_{\mathrm{e}}e^{2}\over 2r_{n}}=-{Z^{2}(k_{\mathrm{e}}e^{2})^{2}m_{% \mathrm{e}}\over 2\hbar^{2}n^{2}}\approx{-13.6Z^{2}\over n^{2}}\mathrm{eV}
  21. R E = ( k e e 2 ) 2 m e 2 2 R_{\mathrm{E}}={(k_{\mathrm{e}}e^{2})^{2}m_{\mathrm{e}}\over 2\hbar^{2}}
  22. m e c 2 \,m_{\mathrm{e}}c^{2}
  23. k e e 2 c = α 1 137 \,{k_{\mathrm{e}}e^{2}\over\hbar c}=\alpha\approx{1\over 137}
  24. R E = 1 2 ( m e c 2 ) α 2 \,R_{\mathrm{E}}={1\over 2}(m_{\mathrm{e}}c^{2})\alpha^{2}
  25. E n = - Z 2 R E n 2 E_{n}=-{Z^{2}R_{\mathrm{E}}\over n^{2}}
  26. m red = m e m p m e + m p = m e 1 1 + m e / m p m\text{red}=\frac{m_{\mathrm{e}}m_{\mathrm{p}}}{m_{\mathrm{e}}+m_{\mathrm{p}}}% =m_{\mathrm{e}}\frac{1}{1+m_{\mathrm{e}}/m_{\mathrm{p}}}
  27. E n = R E 2 n 2 E_{n}={R_{\mathrm{E}}\over 2n^{2}}
  28. E = E i - E f = R E ( 1 n f 2 - 1 n i 2 ) E=E_{i}-E_{f}=R_{\mathrm{E}}\left(\frac{1}{n_{f}^{2}}-\frac{1}{n_{i}^{2}}% \right)\,
  29. E = h c λ , E=\frac{hc}{\lambda},\,
  30. 1 λ = R ( 1 n f 2 - 1 n i 2 ) . \frac{1}{\lambda}=R\left(\frac{1}{n_{f}^{2}}-\frac{1}{n_{i}^{2}}\right).\,
  31. R E / h c R_{\mathrm{E}}/hc
  32. R E / 2 π R_{\mathrm{E}}/2\pi
  33. n f = 1 n_{f}=1
  34. n f = 2 n_{f}=2
  35. n f = 3 n_{f}=3
  36. E = h ν = E i - E f = R E ( Z - 1 ) 2 ( 1 1 2 - 1 2 2 ) E=h\nu=E_{i}-E_{f}=R_{\mathrm{E}}(Z-1)^{2}\left(\frac{1}{1^{2}}-\frac{1}{2^{2}% }\right)\,
  37. f = ν = R v ( 3 4 ) ( Z - 1 ) 2 = ( 2.46 × 10 15 Hz ) ( Z - 1 ) 2 . f=\nu=R_{\mathrm{v}}\left(\frac{3}{4}\right)(Z-1)^{2}=(2.46\times 10^{15}% \operatorname{Hz})(Z-1)^{2}.
  38. 𝐋 = \scriptstyle\mathbf{L}=\hbar
  39. 0 T p r d q r = n h \int_{0}^{T}p_{r}\,dq_{r}=nh\,

Boiling_point.html

  1. T B = ( 1 T 0 - R ln ( P P 0 ) Δ H v a p ) - 1 T_{B}=\Bigg(\frac{1}{T_{0}}-\frac{\,R\,\ln(\frac{P}{P_{0}})}{\Delta H_{vap}}% \Bigg)^{-1}
  2. T B T_{B}
  3. R R
  4. P P
  5. P 0 P_{0}
  6. T 0 T_{0}
  7. Δ H v a p \Delta H_{vap}
  8. P 0 P_{0}
  9. T 0 T_{0}
  10. ln \ln

Boltzmann_constant.html

  1. k = R N A . k=\frac{R}{N\text{A}}.\,
  2. p V = n R T pV=nRT\,
  3. p V = N k T , pV=NkT,
  4. p = 1 3 N V m v 2 ¯ . p=\frac{1}{3}\frac{N}{V}m\overline{v^{2}}.
  5. p V = N k T pV=NkT
  6. 1 2 m v 2 ¯ = 3 2 k T . \tfrac{1}{2}m\overline{v^{2}}=\tfrac{3}{2}kT.
  7. P i exp ( - E k T ) Z , P_{i}\propto\frac{\exp\left(-\frac{E}{kT}\right)}{Z},
  8. S = k ln W . S=k\,\ln W.
  9. Δ S = d Q T . \Delta S=\int\frac{{\rm d}Q}{T}.
  10. S = ln W , Δ S = d Q k T . {S^{\prime}=\ln W},\quad\Delta S^{\prime}=\int\frac{\mathrm{d}Q}{kT}.
  11. V T = k T q , V_{\mathrm{T}}={kT\over q},
  12. E = 1 2 T E=\frac{1}{2}T
  13. S = - P i ln P i . S=-\sum P_{i}\ln P_{i}.

Boltzmann_distribution.html

  1. F ( state ) e - E k T F({\rm state})\propto e^{-\frac{E}{kT}}
  2. E E
  3. k T kT
  4. p i = e - ε i / k T i = 1 M e - ε i / k T p_{i}={\frac{e^{-{\varepsilon}_{i}/kT}}{\sum_{i=1}^{M}{e^{-{\varepsilon}_{i}/% kT}}}}
  5. F ( state2 ) F ( state1 ) = e E 1 - E 2 k T \frac{F({\rm state2})}{F({\rm state1})}=e^{\frac{E_{1}-E_{2}}{kT}}
  6. p i = e - ε i / k T i = 1 M e - ε i / k T p_{i}={\frac{e^{-{\varepsilon}_{i}/kT}}{\sum_{i=1}^{M}{e^{-{\varepsilon}_{i}/% kT}}}}
  7. Q = i = 1 M e - ε i / k T Q={\sum_{i=1}^{M}{e^{-{\varepsilon}_{i}/kT}}}
  8. p i = 1 Q e - ε i / k T p_{i}={\frac{1}{Q}}{e^{-{\varepsilon}_{i}/kT}}
  9. p i p j = e ( ε j - ε i ) / k T {\frac{p_{i}}{p_{j}}}=e^{({\varepsilon}_{j}-{\varepsilon}_{i})/kT}
  10. p i = N i N p_{i}={\frac{N_{i}}{N}}
  11. N i N = e - ε i / k T i = 1 M e - ε i / k T {\frac{N_{i}}{N}}={\frac{e^{-{\varepsilon}_{i}/kT}}{\sum_{i=1}^{M}{e^{-{% \varepsilon}_{i}/kT}}}}

Bolzano–Weierstrass_theorem.html

  1. x n 2 x n 1 . x_{n_{2}}\geq x_{n_{1}}.
  2. x n 3 x n 2 . x_{n_{3}}\geq x_{n_{2}}.
  3. x n 1 x n 2 x n 3 x_{n_{1}}\leq x_{n_{2}}\leq x_{n_{3}}\leq\ldots

Boolean_satisfiability_problem.html

  1. O ( 2 0.386 n ) O(2^{0.386n})

Borel_measure.html

  1. 𝔅 ( X ) \mathfrak{B}(X)
  2. 𝔅 ( ) \mathfrak{B}(\mathbb{R})
  3. \mathbb{R}
  4. μ ( [ a , b ] ) = b - a \mu([a,b])=b-a
  5. [ a , b ] [a,b]
  6. \mathbb{R}
  7. λ \lambda
  8. λ \lambda
  9. μ \mu
  10. λ ( E ) = μ ( E ) \lambda(E)=\mu(E)
  11. ( μ ) ( s ) = [ 0 , ) e - s t d μ ( t ) . (\mathcal{L}\mu)(s)=\int_{[0,\infty)}e^{-st}\,d\mu(t).
  12. ( f ) ( s ) = 0 - e - s t f ( t ) d t (\mathcal{L}f)(s)=\int_{0^{-}}^{\infty}e^{-st}f(t)\,dt
  13. lim ε 0 - ε . \lim_{\varepsilon\downarrow 0}\int_{-\varepsilon}^{\infty}.
  14. μ ( B ( x , r ) ) r s \mu(B(x,r))\leq r^{s}
  15. R k R^{k}

Borel_set.html

  1. T σ T_{\sigma}\quad
  2. T δ T_{\delta}\quad
  3. T δ σ = ( T δ ) σ . T_{\delta\sigma}=(T_{\delta})_{\sigma}.\,
  4. G 0 G^{0}
  5. G i = [ G i - 1 ] δ σ . G^{i}=[G^{i-1}]_{\delta\sigma}.
  6. G i = j < i G j . G^{i}=\bigcup_{j<i}G^{j}.
  7. G G δ σ . G\mapsto G_{\delta\sigma}.
  8. 1 × 2 0 = 2 0 . \aleph_{1}\times 2^{\aleph_{0}}\,=2^{\aleph_{0}}.\,
  9. f : X Y f:X\rightarrow Y
  10. f - 1 ( B ) f^{-1}(B)
  11. x = a 0 + 1 a 1 + 1 a 2 + 1 a 3 + 1 x=a_{0}+\cfrac{1}{a_{1}+\cfrac{1}{a_{2}+\cfrac{1}{a_{3}+\cfrac{1}{\ddots\,}}}}
  12. a 0 a_{0}\,
  13. a k a_{k}\,
  14. A A\,
  15. ( a 0 , a 1 , ) (a_{0},a_{1},\dots)\,
  16. ( a k 0 , a k 1 , ) (a_{k_{0}},a_{k_{1}},\dots)\,
  17. A A\,
  18. f - 1 [ 0 ] f^{-1}[0]
  19. f : { 0 , 1 } ω { 0 , 1 } f\colon\{0,1\}^{\omega}\to\{0,1\}
  20. X X
  21. σ \sigma
  22. X X
  23. X X
  24. X X

Borel–Cantelli_lemma.html

  1. n = 1 Pr ( E n ) < , \sum_{n=1}^{\infty}\Pr(E_{n})<\infty,
  2. Pr ( lim sup n E n ) = 0. \Pr\left(\limsup_{n\to\infty}E_{n}\right)=0.\,
  3. lim sup n E n = n = 1 k = n E k . \limsup_{n\to\infty}E_{n}=\bigcap_{n=1}^{\infty}\bigcup_{k=n}^{\infty}E_{k}.
  4. E 0 , E 1 , E 2 , E_{0},E_{1},E_{2},\ldots
  5. N M = def m = 0 M - 1 [ E m ] N_{M}\stackrel{\,\text{def}}{=}\sum_{m=0}^{M-1}[E_{m}]
  6. E E
  7. [ E ] [E]
  8. E E
  9. [ E ] = def { 1 E 0 otherwise [E]\stackrel{\,\text{def}}{=}\begin{cases}1&E\\ 0&\,\text{otherwise}\end{cases}
  10. N M N_{M}
  11. m < M m<M
  12. E m E_{m}
  13. Pr ( N M > N ) 1 N E [ N M ] \Pr(N_{M}>N)\leq\frac{1}{N}\operatorname{E}[N_{M}]
  14. X X
  15. E [ X ] \operatorname{E}[X]
  16. X X
  17. E [ N M ] = m = 0 M - 1 Pr ( E m ) \operatorname{E}[N_{M}]=\sum_{m=0}^{M-1}\Pr(E_{m})
  18. M M
  19. sup M Pr ( N M > N ) 1 N sup M E [ N M ] \sup_{M}\Pr(N_{M}>N)\leq\frac{1}{N}\sup_{M}\operatorname{E}[N_{M}]
  20. N ω = def sup M N M N_{\omega}\stackrel{\,\text{def}}{=}\sup_{M}N_{M}
  21. N 0 , N 1 , N 2 , N_{0},N_{1},N_{2},\ldots
  22. N i N j N_{i}\leq N_{j}
  23. i < j i<j
  24. [ N 0 > N ] , [ N 1 > N ] , [ N 2 > N ] , [N_{0}>N],[N_{1}>N],[N_{2}>N],\ldots
  25. Pr ( N ω > N ) sup M Pr ( N M > N ) \Pr(N_{\omega}>N)\leq\sup_{M}\Pr(N_{M}>N)
  26. Pr ( N ω > N ) 1 N E [ N ω ] \Pr(N_{\omega}>N)\leq\frac{1}{N}\operatorname{E}[N_{\omega}]
  27. E [ N ω ] \operatorname{E}[N_{\omega}]
  28. N N
  29. inf N Pr ( N ω > N ) 0 \inf_{N}\Pr(N_{\omega}>N)\leq 0
  30. N ( N ω > N ) N ω > N \forall_{N}(N_{\omega}>N)\Rightarrow N_{\omega}>N
  31. Pr [ N ( N ω > N ) ] inf N Pr ( N ω > N ) \Pr[\forall_{N}(N_{\omega}>N)]\leq\inf_{N}\Pr(N_{\omega}>N)
  32. Pr [ N ( N ω > N ) ] 0 \Pr[\forall_{N}(N_{\omega}>N)]\leq 0
  33. N ( N ω > N ) lim sup m E m \forall_{N}(N_{\omega}>N)\Leftrightarrow\limsup_{m}E_{m}
  34. n = 1 Pr ( E n ) < . \sum_{n=1}^{\infty}\Pr(E_{n})<\infty.
  35. Pr ( E i ) Pr ( E i + 1 ) . \Pr(E_{i})\geq\Pr(E_{i+1}).
  36. n = N Pr ( E n ) 0 \sum_{n=N}^{\infty}\Pr(E_{n})\rightarrow 0
  37. N N
  38. inf N 1 n = N Pr ( E n ) = 0. \inf_{N\geq 1}\sum_{n=N}^{\infty}\Pr(E_{n})=0.\,
  39. Pr ( lim sup n E n ) = Pr ( infinitely many of the E n occur ) = Pr ( N = 1 n = N E n ) inf N 1 Pr ( n = N E n ) inf N 1 n = N Pr ( E n ) = 0. \begin{aligned}&\displaystyle{}\qquad\Pr\left(\limsup_{n\to\infty}E_{n}\right)% =\Pr(\,\text{infinitely many of the }E_{n}\,\text{ occur})\\ &\displaystyle=\Pr\left(\bigcap_{N=1}^{\infty}\bigcup_{n=N}^{\infty}E_{n}% \right)\leq\inf_{N\geq 1}\Pr\left(\bigcup_{n=N}^{\infty}E_{n}\right)\leq\inf_{% N\geq 1}\sum_{n=N}^{\infty}\Pr(E_{n})=0.\end{aligned}
  40. n = 1 μ ( A n ) < , \sum_{n=1}^{\infty}\mu(A_{n})<\infty,
  41. μ ( lim sup n A n ) = 0. \mu\left(\limsup_{n\to\infty}A_{n}\right)=0.\,
  42. n = 1 Pr ( E n ) = \sum^{\infty}_{n=1}\Pr(E_{n})=\infty
  43. ( E n ) n = 1 (E_{n})^{\infty}_{n=1}
  44. Pr ( lim sup n E n ) = 1. \Pr(\limsup_{n\rightarrow\infty}E_{n})=1.
  45. j μ ( E j ) = , \sum_{j}\mu(E_{j})=\infty,
  46. F j = E j + x j F_{j}=E_{j}+x_{j}\,
  47. lim sup F j = n = 1 k = n F k = n \lim\sup F_{j}=\bigcap_{n=1}^{\infty}\bigcup_{k=n}^{\infty}F_{k}=\mathbb{R}^{n}
  48. n = 1 Pr ( E n ) = \sum_{n=1}^{\infty}\Pr(E_{n})=\infty
  49. ( E n ) n = 1 (E_{n})^{\infty}_{n=1}
  50. 1 - Pr ( lim sup n E n ) = 0. 1-\Pr(\limsup_{n\rightarrow\infty}E_{n})=0.\,
  51. 1 - Pr ( lim sup n E n ) \displaystyle 1-\Pr(\limsup_{n\rightarrow\infty}E_{n})
  52. Pr ( n = N E n c ) = 0 \Pr\left(\bigcap_{n=N}^{\infty}E_{n}^{c}\right)=0
  53. ( E n ) n = 1 (E_{n})^{\infty}_{n=1}
  54. Pr ( n = N E n c ) \displaystyle\Pr\left(\bigcap_{n=N}^{\infty}E_{n}^{c}\right)
  55. Pr ( n = N E n c ) = 0 \Pr\left(\bigcap_{n=N}^{\infty}E_{n}^{c}\right)=0
  56. - log ( Pr ( n = N E n c ) ) \displaystyle-\log\left(\Pr\left(\bigcap_{n=N}^{\infty}E_{n}^{c}\right)\right)
  57. n = 1 Pr ( E n ) = . \sum^{\infty}_{n=1}\Pr(E_{n})=\infty.
  58. ( A n ) (A_{n})
  59. ( A n ) (A_{n})
  60. A k A k + 1 A_{k}\subseteq A_{k+1}
  61. A ¯ \bar{A}
  62. A A
  63. A k A_{k}
  64. A k A_{k}
  65. ( t k ) (t_{k})
  66. k Pr ( A t k + 1 A ¯ t k ) = . \sum_{k}\Pr(A_{t_{k+1}}\mid\bar{A}_{t_{k}})=\infty.
  67. ( t k ) (t_{k})

Born–Oppenheimer_approximation.html

  1. Ψ total = ψ electronic × ψ nuclear \Psi_{\mathrm{total}}=\psi_{\mathrm{electronic}}\times\psi_{\mathrm{nuclear}}
  2. ψ electronic \psi_{\,\text{electronic}}
  3. E total = E electronic + E vibrational + E rotational + E nuclear E_{\mathrm{total}}=E_{\mathrm{electronic}}+E_{\mathrm{vibrational}}+E_{\mathrm% {rotational}}+E_{\mathrm{nuclear}}
  4. H e ( 𝐫 , 𝐑 ) χ ( 𝐫 , 𝐑 ) = E e χ ( 𝐫 , 𝐑 ) H_{\mathrm{e}}(\mathbf{r,R})\;\chi(\mathbf{r,R})=E_{\mathrm{e}}\;\chi(\mathbf{% r,R})
  5. [ T n + E e ( 𝐑 ) ] ϕ ( 𝐑 ) = E ϕ ( 𝐑 ) \left[T_{\mathrm{n}}+E_{\mathrm{e}}(\mathbf{R})\right]\phi(\mathbf{R})=E\phi(% \mathbf{R})
  6. E 0 ( 𝐑 ) E 1 ( 𝐑 ) E 2 ( 𝐑 ) for all 𝐑 E_{0}(\mathbf{R})\ll E_{1}(\mathbf{R})\ll E_{2}(\mathbf{R})\ll\cdots\,\text{ % for all }\mathbf{R}
  7. H = H e + T n H=H_{\mathrm{e}}+T_{\mathrm{n}}\,
  8. H e = - i 1 2 i 2 - i , A Z A r i A + i > j 1 r i j + A > B Z A Z B R A B and T n = - A 1 2 M A A 2 . H_{\mathrm{e}}=-\sum_{i}{\frac{1}{2}\nabla_{i}^{2}}-\sum_{i,A}{\frac{Z_{A}}{r_% {iA}}}+\sum_{i>j}{\frac{1}{r_{ij}}}+\sum_{A>B}{\frac{Z_{A}Z_{B}}{R_{AB}}}\quad% \mathrm{and}\quad T_{\mathrm{n}}=-\sum_{A}{\frac{1}{2M_{A}}\nabla_{A}^{2}}.
  9. 𝐫 { 𝐫 i } \mathbf{r}\equiv\{\mathbf{r}_{i}\}
  10. 𝐑 { 𝐑 A = ( R A x , R A y , R A z ) } \mathbf{R}\equiv\{\mathbf{R}_{A}=(R_{Ax},\,R_{Ay},\,R_{Az})\}
  11. r i A | 𝐫 i - 𝐑 A | r_{iA}\equiv|\mathbf{r}_{i}-\mathbf{R}_{A}|
  12. r i j r_{ij}\;
  13. R A B R_{AB}\,
  14. T n = A α = x , y , z P A α P A α 2 M A with P A α = - i R A α . T_{\mathrm{n}}=\sum_{A}\sum_{\alpha=x,y,z}\frac{P_{A\alpha}P_{A\alpha}}{2M_{A}% }\quad\mathrm{with}\quad P_{A\alpha}=-i{\partial\over\partial R_{A\alpha}}.
  15. χ k ( 𝐫 ; 𝐑 ) \chi_{k}(\mathbf{r};\mathbf{R})
  16. H e H_{\mathrm{e}}\,
  17. H e χ k ( 𝐫 ; 𝐑 ) = E k ( 𝐑 ) χ k ( 𝐫 ; 𝐑 ) for k = 1 , , K . H_{\mathrm{e}}\;\chi_{k}(\mathbf{r};\mathbf{R})=E_{k}(\mathbf{R})\;\chi_{k}(% \mathbf{r};\mathbf{R})\quad\mathrm{for}\quad k=1,\ldots,K.
  18. χ k \chi_{k}\,
  19. χ k \chi_{k}\,
  20. χ k \chi_{k}\,
  21. 𝐫 \mathbf{r}
  22. 𝐑 \mathbf{R}
  23. χ k \chi_{k}\,
  24. 𝐑 \mathbf{R}
  25. χ k \chi_{k}\,
  26. P A α χ k ( 𝐫 ; 𝐑 ) = - i χ k ( 𝐫 ; 𝐑 ) R A α for α = x , y , z , P_{A\alpha}\chi_{k}(\mathbf{r};\mathbf{R})=-i\frac{\partial\chi_{k}(\mathbf{r}% ;\mathbf{R})}{\partial R_{A\alpha}}\quad\mathrm{for}\quad\alpha=x,y,z,
  27. Ψ ( 𝐑 , 𝐫 ) \Psi(\mathbf{R},\mathbf{r})
  28. χ k ( 𝐫 ; 𝐑 ) \chi_{k}(\mathbf{r};\mathbf{R})
  29. Ψ ( 𝐑 , 𝐫 ) = k = 1 K χ k ( 𝐫 ; 𝐑 ) ϕ k ( 𝐑 ) , \Psi(\mathbf{R},\mathbf{r})=\sum_{k=1}^{K}\chi_{k}(\mathbf{r};\mathbf{R})\phi_% {k}(\mathbf{R}),
  30. χ k ( 𝐫 ; 𝐑 ) | χ k ( 𝐫 ; 𝐑 ) ( 𝐫 ) = δ k k \langle\,\chi_{k^{\prime}}(\mathbf{r};\mathbf{R})\,|\,\chi_{k}(\mathbf{r};% \mathbf{R})\rangle_{(\mathbf{r})}=\delta_{k^{\prime}k}
  31. ( 𝐫 ) (\mathbf{r})
  32. ( e ( 𝐑 ) ) k k χ k ( 𝐫 ; 𝐑 ) | H e | χ k ( 𝐫 ; 𝐑 ) ( 𝐫 ) = δ k k E k ( 𝐑 ) \big(\mathbb{H}_{\mathrm{e}}(\mathbf{R})\big)_{k^{\prime}k}\equiv\langle\chi_{% k^{\prime}}(\mathbf{r};\mathbf{R})|H_{\mathrm{e}}|\chi_{k}(\mathbf{r};\mathbf{% R})\rangle_{(\mathbf{r})}=\delta_{k^{\prime}k}E_{k}(\mathbf{R})
  33. χ k ( 𝐫 ; 𝐑 ) \chi_{k^{\prime}}(\mathbf{r};\mathbf{R})
  34. 𝐫 \mathbf{r}
  35. H Ψ ( 𝐑 , 𝐫 ) = E Ψ ( 𝐑 , 𝐫 ) H\;\Psi(\mathbf{R},\mathbf{r})=E\;\Psi(\mathbf{R},\mathbf{r})
  36. [ n ( 𝐑 ) + e ( 𝐑 ) ] s y m b o l ϕ ( 𝐑 ) = E s y m b o l ϕ ( 𝐑 ) . \left[\mathbb{H}_{\mathrm{n}}(\mathbf{R})+\mathbb{H}_{\mathrm{e}}(\mathbf{R})% \right]\;symbol{\phi}(\mathbf{R})=E\;symbol{\phi}(\mathbf{R}).
  37. s y m b o l ϕ ( 𝐑 ) symbol{\phi}(\mathbf{R})
  38. ϕ k ( 𝐑 ) , k = 1 , , K \phi_{k}(\mathbf{R}),\;k=1,\ldots,K
  39. e ( 𝐑 ) \mathbb{H}_{\mathrm{e}}(\mathbf{R})
  40. ( n ( 𝐑 ) ) k k = χ k ( 𝐫 ; 𝐑 ) | T n | χ k ( 𝐫 ; 𝐑 ) ( 𝐫 ) . \big(\mathbb{H}_{\mathrm{n}}(\mathbf{R})\big)_{k^{\prime}k}=\langle\chi_{k^{% \prime}}(\mathbf{r};\mathbf{R})|T_{\mathrm{n}}|\chi_{k}(\mathbf{r};\mathbf{R})% \rangle_{(\mathbf{r})}.
  41. T n T_{\textrm{n}}
  42. H n ( 𝐑 ) k k ( n ( 𝐑 ) ) k k = δ k k T n + A , α 1 M A χ k | ( P A α χ k ) ( 𝐫 ) P A α + χ k | ( T n χ k ) ( 𝐫 ) . \mathrm{H_{n}}(\mathbf{R})_{k^{\prime}k}\equiv\big(\mathbb{H}_{\mathrm{n}}(% \mathbf{R})\big)_{k^{\prime}k}=\delta_{k^{\prime}k}T_{\textrm{n}}+\sum_{A,% \alpha}\frac{1}{M_{A}}\langle\chi_{k^{\prime}}|\big(P_{A\alpha}\chi_{k}\big)% \rangle_{(\mathbf{r})}P_{A\alpha}+\langle\chi_{k^{\prime}}|\big(T_{\mathrm{n}}% \chi_{k}\big)\rangle_{(\mathbf{r})}.
  43. k = k k^{\prime}=k
  44. χ k | ( P A α χ k ) ( 𝐫 ) \langle\chi_{k}|\big(P_{A\alpha}\chi_{k}\big)\rangle_{(\mathbf{r})}
  45. P A α P_{A\alpha}\,
  46. χ k \chi_{k}
  47. χ k | ( P A α χ k ) ( 𝐫 ) = χ k | [ P A α , H e ] | χ k ( 𝐫 ) E k ( 𝐑 ) - E k ( 𝐑 ) . \langle\chi_{k^{\prime}}|\big(P_{A\alpha}\chi_{k}\big)\rangle_{(\mathbf{r})}=% \frac{\langle\chi_{k^{\prime}}|\big[P_{A\alpha},H_{\mathrm{e}}\big]|\chi_{k}% \rangle_{(\mathbf{r})}}{E_{k}(\mathbf{R})-E_{k^{\prime}}(\mathbf{R})}.
  48. χ k | [ P A α , H e ] | χ k ( 𝐫 ) = i Z A i χ k | ( 𝐫 i A ) α r i A 3 | χ k ( 𝐫 ) with 𝐫 i A 𝐫 i - 𝐑 A . \langle\chi_{k^{\prime}}|\big[P_{A\alpha},H_{\mathrm{e}}\big]|\chi_{k}\rangle_% {(\mathbf{r})}=iZ_{A}\sum_{i}\;\langle\chi_{k^{\prime}}|\frac{(\mathbf{r}_{iA}% )_{\alpha}}{r_{iA}^{3}}|\chi_{k}\rangle_{(\mathbf{r})}\;\;\mathrm{with}\;\;% \mathbf{r}_{iA}\equiv\mathbf{r}_{i}-\mathbf{R}_{A}.
  49. E k ( 𝐑 ) E k ( 𝐑 ) {E_{k}(\mathbf{R})\approx E_{k^{\prime}}(\mathbf{R})}
  50. P α A P^{A}_{\alpha}
  51. P α A P^{A}_{\alpha}
  52. [ T n + E k ( 𝐑 ) ] ϕ k ( 𝐑 ) = E ϕ k ( 𝐑 ) for k = 1 , , K , \left[T_{\mathrm{n}}+E_{k}(\mathbf{R})\right]\;\phi_{k}(\mathbf{R})=E\phi_{k}(% \mathbf{R})\quad\mathrm{for}\quad k=1,\ldots,K,
  53. 𝐪 \mathbf{q}
  54. u 1 ( 𝐪 ) u_{1}(\mathbf{q})
  55. u 2 ( 𝐪 ) u_{2}(\mathbf{q})
  56. u 2 ( 𝐪 ) u_{2}(\mathbf{q})
  57. u 1 ( 𝐪 ) {u}_{1}(\mathbf{q})
  58. u 2 ( 𝐪 ) {u}_{2}(\mathbf{q})
  59. 2 2 m ( + τ ) 2 Ψ + ( 𝐮 - E ) Ψ = 0 \frac{\hbar^{2}}{2m}(\nabla+\tau)^{2}\Psi+(\mathbf{u}-E)\Psi=0
  60. Ψ ( 𝐪 ) \Psi(\mathbf{q})
  61. ψ k ( 𝐪 ) \psi_{k}(\mathbf{q})
  62. 𝐮 ( 𝐪 ) \mathbf{u}(\mathbf{q})
  63. u k ( 𝐪 ) u_{k}(\mathbf{q})
  64. \nabla
  65. 𝐪 \mathbf{q}
  66. τ ( 𝐪 ) \mathbf{\tau}(\mathbf{q})
  67. τ j k = ζ j | ζ k \mathbf{\tau}_{jk}=\langle\zeta_{j}|\nabla\zeta_{k}\rangle
  68. | ζ n ; n = j , k |\zeta_{n}\rangle;n=j,k
  69. - 2 2 m 2 ψ 1 + ( u ~ 1 - E ) ψ 1 - 2 2 m [ 2 τ 12 + τ 12 ] ψ 2 = 0 -\frac{\hbar^{2}}{2m}\nabla^{2}\psi_{1}+(\tilde{u}_{1}-E)\psi_{1}-\frac{\hbar^% {2}}{2m}[2\mathbf{\tau}_{12}\nabla+\nabla\mathbf{\tau}_{12}]\psi_{2}=0
  70. - 2 2 m 2 ψ 2 + ( u ~ 2 - E ) ψ 2 + 2 2 m [ 2 τ 12 + τ 12 ] ψ 1 = 0 -\frac{\hbar^{2}}{2m}\nabla^{2}\psi_{2}+(\tilde{u}_{2}-E)\psi_{2}+\frac{\hbar^% {2}}{2m}[2\mathbf{\tau}_{12}\nabla+\nabla\mathbf{\tau}_{12}]\psi_{1}=0
  71. u ~ k ( 𝐪 ) = u k ( 𝐪 ) + ( 2 / 2 m ) τ 12 2 ; k = 1 , 2 , \tilde{u}_{k}(\mathbf{q})=u_{k}(\mathbf{q})+(\hbar^{2}/2m)\tau_{12}^{2};k=1,2,
  72. τ 12 ( = τ 12 ( 𝐪 ) ) \mathbf{\tau}_{12}(=\mathbf{\tau}_{12}(\mathbf{q}))
  73. u 1 ( 𝐪 ) u_{1}(\mathbf{q})
  74. u 2 ( 𝐪 ) u_{2}(\mathbf{q})
  75. χ = ψ 1 + i ψ 2 \chi=\psi_{1}+i\psi_{2}\,
  76. i i
  77. - 2 2 m 2 χ + ( u ~ 1 - E ) χ + i 2 2 m [ 2 τ 12 + τ 12 ] χ + i ( u 1 - u 2 ) ψ 2 = 0 -\frac{\hbar^{2}}{2m}\nabla^{2}\chi+(\tilde{u}_{1}-E)\chi+i\frac{\hbar^{2}}{2m% }[2\mathbf{\tau}_{12}\nabla+\nabla\mathbf{\tau}_{12}]\chi+i({u}_{1}-{u}_{2})% \psi_{2}=0
  78. u 2 ( 𝐪 ) {u}_{2}(\mathbf{q})
  79. ψ 2 ( 𝐪 ) \psi_{2}(\mathbf{q})
  80. u 2 ( 𝐪 ) {u}_{2}(\mathbf{q})
  81. u 1 ( 𝐪 ) {u}_{1}(\mathbf{q})
  82. u 2 ( 𝐪 ) {u}_{2}(\mathbf{q})
  83. u 1 ( 𝐪 ) {u}_{1}(\mathbf{q})
  84. u 2 ( 𝐪 ) {u}_{2}(\mathbf{q})
  85. - 2 2 m 2 χ + ( u ~ 1 - E ) χ + i 2 2 m [ 2 τ 12 + τ 12 ] χ = 0 -\frac{\hbar^{2}}{2m}\nabla^{2}\chi+(\tilde{u}_{1}-E)\chi+i\frac{\hbar^{2}}{2m% }[2\mathbf{\tau}_{12}\nabla+\nabla\mathbf{\tau}_{12}]\chi=0
  86. u 0 ( 𝐪 ) {u}_{0}(\mathbf{q})
  87. u 1 ( 𝐪 ) {u}_{1}(\mathbf{q})
  88. χ 0 \chi_{0}
  89. χ 0 ( 𝐪 | Γ ) = ξ 0 ( 𝐪 ) e x p [ - i Γ d 𝐪 τ ( 𝐪 | Γ ) ] \chi_{0}(\mathbf{q}|\Gamma)=\xi_{0}(\mathbf{q})exp[-i\int_{\Gamma}d\mathbf{q}% \mathbf{{}^{\prime}}\cdot\mathbf{\tau}(\mathbf{q}\mathbf{{}^{\prime}}|\Gamma)]
  90. Γ \Gamma
  91. Γ \Gamma
  92. ξ 0 ( 𝐪 ) \xi_{0}(\mathbf{q})
  93. - 2 2 m 2 ξ 0 + ( u 0 - E ) ξ 0 = 0 -\frac{\hbar^{2}}{2m}\nabla^{2}\xi_{0}+(u_{0}-E)\xi_{0}=0
  94. χ 0 ( 𝐪 | Γ ) \chi_{0}(\mathbf{q}|\Gamma)
  95. χ ( 𝐪 | Γ ) = χ 0 ( 𝐪 | Γ ) + η ( 𝐪 | Γ ) \chi(\mathbf{q}|\Gamma)=\chi_{0}(\mathbf{q}|\Gamma)+\eta(\mathbf{q}|\Gamma)
  96. η ( 𝐪 | Γ ) \eta(\mathbf{q}|\Gamma)
  97. - 2 2 m 2 η + ( u ~ 1 - E ) η + i 2 2 m [ 2 τ 12 + τ 12 ] η = ( u 1 - u 0 ) χ 0 -\frac{\hbar^{2}}{2m}\nabla^{2}\eta+(\tilde{u}_{1}-E)\eta+i\frac{\hbar^{2}}{2m% }[2\mathbf{\tau}_{12}\nabla+\nabla\mathbf{\tau}_{12}]\eta=({u}_{1}-{u}_{0})% \chi_{0}

Borsuk–Ulam_theorem.html

  1. f : S n R n f:S^{n}\to R^{n}
  2. x S n x\in S^{n}
  3. f ( - x ) = f ( x ) f(-x)=f(x)
  4. n = 1 n=1
  5. n = 2 n=2
  6. S n S^{n}
  7. B n B^{n}
  8. g : S n R n g:S^{n}\to R^{n}
  9. x S n x\in S^{n}
  10. g ( x ) = 0 g(x)=0
  11. g : B n R n g:B^{n}\to R^{n}
  12. S n - 1 S^{n-1}
  13. B n B^{n}
  14. x B n x\in B^{n}
  15. g ( x ) = 0 g(x)=0
  16. g g
  17. x x
  18. g ( - x ) = - g ( x ) g(-x)=-g(x)
  19. g ( - x ) = g ( x ) g(-x)=g(x)
  20. g ( x ) = 0 g(x)=0
  21. f f
  22. g ( x ) = f ( x ) - f ( - x ) g(x)=f(x)-f(-x)
  23. g g
  24. f ( x ) = f ( - x ) f(x)=f(-x)
  25. h : S n S n - 1 h:S^{n}\to S^{n-1}
  26. S n S^{n}
  27. 0 S n - 1 0\notin S^{n-1}
  28. S n - 1 S^{n-1}
  29. g : S n R n g:S^{n}\to R^{n}
  30. h : S n S n - 1 h:S^{n}\to S^{n-1}
  31. h ( x ) = g ( x ) | g ( x ) | h(x)=\frac{g(x)}{|g(x)|}
  32. g g
  33. h h
  34. g g
  35. x x
  36. g ( x ) = 0 g(x)=0
  37. g ( x ) > 0 g(x)>0
  38. g ( - x ) < 0 g(-x)<0
  39. y y
  40. x x
  41. - x -x
  42. g ( y ) = 0 g(y)=0
  43. h * : S n - 1 S n - 1 h^{*}:S^{n-1}\to S^{n-1}
  44. h : S n S n - 1 h:S^{n}\to S^{n-1}
  45. h * : S n - 1 S n - 1 h^{*}:S^{n-1}\to S^{n-1}
  46. h h
  47. h * h^{*}
  48. h * h^{*}
  49. S n S^{n}
  50. h h
  51. g : S n R n g:S^{n}\to R^{n}
  52. ϵ > 0 \epsilon>0
  53. δ > 0 \delta>0
  54. S n S_{n}
  55. δ \delta
  56. ϵ \epsilon
  57. S n S_{n}
  58. δ \delta
  59. v v
  60. l ( v ) ± 1 , ± 2 , , ± n l(v)\in{\pm 1,\pm 2,...,\pm n}
  61. | l ( v ) | = arg max k ( g ( v ) k ) |l(v)|=\arg\max_{k}(g(v)_{k})
  62. l ( v ) = sgn ( g ( v ) ) | l ( v ) | l(v)=\operatorname{sgn}(g(v))|l(v)|
  63. l ( - v ) = - l ( v ) l(-v)=-l(v)
  64. u , v u,v
  65. l ( u ) = 1 , l ( v ) = - 1 l(u)=1,l(v)=-1
  66. g ( u ) g(u)
  67. g ( v ) g(v)
  68. g ( u ) g(u)
  69. g ( v ) g(v)
  70. g ( u ) g(u)
  71. g ( v ) g(v)
  72. ϵ \epsilon
  73. | g ( u ) | |g(u)|
  74. | g ( v ) | |g(v)|
  75. ϵ \epsilon
  76. ϵ \epsilon
  77. | g ( u ) | = 0 |g(u)|=0