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Complex Function 复变函数

Curve graph 曲线图

List of Complex Functions 复变函数目录

Hyperlinks lead to plots in two dimensions of the real and imaginary parts of functions on the real and imaginary axes, as well as visualizations in three dimensions of the real and imaginary parts and their absolute value on the complex plane. The 3D graph can be zoom and rotated with mouse wheel.

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    初等复变函数

    Basic Functions 基本初等函数

  1. abs( x ) — absolute value of a real or complex number
  2. arg( x ) — argument of a real or complex number
  3. pow( x, y ) — power of a real or complex number to a real or complex exponent
  4. root( x, y ) — root of a real or complex number with real or complex degree
  5. sqrt( x ) — square root of a real or complex number
  6. cbrt( x ) — cubic root of a real or complex number

  7. Logarithmic Functions 对数函数

  8. exp( x ) — exponential of a real or complex number
  9. ln(x)=log( x ) — natural logarithm of a real or complex number
  10. log( x, base ) — logarithm of a real or complex number to a real or complex base
  11. lambertW( x ) — principal branch of the Lambert W-function of a real number
  12. lambertW( k, x ) — real branches of the Lambert W-function of a real number for k = −1 or k = 0

  13. Circular Functions 三角函数

  14. sin( x ) — sine of a real or complex number
  15. cos( x ) — cosine of a real or complex number
  16. tan( x ) — tangent of a real or complex number
  17. cot( x ) — cotangent of a real or complex number
  18. sec( x ) — secant of a real or complex number
  19. csc( x ) — cosecant of a real or complex number
  20. asin(x)=arcsin( x ) — inverse sine of a real or complex number
  21. acos(x)=arccos( x ) — inverse cosine of a real or complex number
  22. atan(x)=arctan( x ) — inverse tangent of a real or complex number
  23. acot(x)=arccot( x ) — inverse cotangent of a real or complex number
  24. asec(x)=arcsec( x ) — inverse secant of a real or complex number
  25. acsc(x)=arccsc( x ) — inverse cosecant of a real or complex number
  26. sinc( x ) — cardinal sine of a real or complex number

  27. Hyperbolic Functions 双曲函数

  28. sinh( x ) — hyperbolic sine of a real or complex number
  29. cosh( x ) — hyperbolic cosine of a real or complex number
  30. tanh( x ) — hyperbolic tangent of a real or complex number
  31. coth( x ) — hyperbolic cotangent of a real or complex number
  32. sech( x ) — hyperbolic secant of a real or complex number
  33. csch( x ) — hyperbolic cosecant of a real or complex number
  34. asinh(x)=arcsinh( x ) — inverse hyperbolic sine of a real or complex number
  35. acosh(x)=arccosh( x ) — inverse hyperbolic cosine of a real or complex number
  36. atanh(x)=arctanh( x ) — inverse hyperbolic tangent of a real or complex number
  37. acoth(x)=arccoth( x ) — inverse hyperbolic cotangent of a real or complex number
  38. asech(x)=arcsech( x ) — inverse secant of a real or complex number
  39. acsch(x)=arccsch( x ) — inverse hyperbolic cosecant of a real or complex number
  40. gudermannian( x ) — Gudermannian function of a real or complex number, = arctan( sinh(x) )
  41. inverseGudermannian( x ) — inverse Gudermannian function of a real or complex number, = arctanh( sin(x) )

  42. Special Function 特殊函数


    Bessel Functions 贝塞耳函数

  43. besselJ( n, x ) — Bessel function of the first kind of real or complex order n of a real or complex number
  44. besselJZero( n, m )mth zero of the Bessel function of the first kind of positive order n
  45. besselJZero( n, m, true )mth zero of the first derivative of the Bessel function of the first kind of positive order n
  46. besselY( n, x ) — Bessel function of the second kind of real or complex order n of a real or complex number
  47. besselYZero( n, m )mth zero of the Bessel function of the second kind of positive order n
  48. besselYZero( n, m, true )mth zero of the first derivative of the Bessel function of the second kind of positive order n
  49. besselI( n, x ) — modified Bessel function of the first kind of real or complex order n of a real or complex number
  50. besselK( n, x ) — modified Bessel function of the second kind of real or complex order n of a real or complex number
  51. hankel1( n, x ) — Hankel function of the first kind of real or complex order n of a real or complex number
  52. hankel2( n, x ) — Hankel function of the second kind of real or complex order n of a real or complex number

  53. Bessel-Type Functions

  54. Ai(x)=airyAi( x ) — Airy function of the first kind of a real or complex number
  55. AiPrime(x)=airyAiPrime( x ) — derivative of the Airy function of the first kind of a real or complex number
  56. Bi(x)=airyBi( x ) — Airy function of the second kind of a real or complex number
  57. BiPrime(x)=airyBiPrime( x ) — derivative of the Airy function of the second kind of a real or complex number
  58. sphericalBesselJ( n, x ) — spherical Bessel function of the first kind of real or complex order n of a real or complex number
  59. sphericalBesselY( n, x ) — spherical Bessel function of the second kind of real or complex order n of a real or complex number
  60. sphericalHankel1( n, x ) — spherical Hankel function of the first kind of real or complex order n of a real or complex number
  61. sphericalHankel2( n, x ) — spherical Hankel function of the second kind of real or complex order n of a real or complex number
  62. struveH( n, x ) — Struve function of real or complex order n of a real or complex number
  63. struveL( n, x ) — modified Struve function of real or complex order n of a real or complex number

  64. Orthogonal Polynomials 正交多项式

  65. hermite( n, x ) — Hermite polynomial of real or complex index n of a real or complex number
  66. laguerre( n, x ) — Laguerre polynomial of real or complex index n of a real or complex number
  67. laguerre( n, a, x ) — associated Laguerre polynomial of real or complex index n and real or complex argument a of a real or complex number
  68. legendreP( l, x ) — Legendre polynomial of real or complex index l of a real or complex number
  69. legendreP( l, m, x ) — associated Legendre polynomial of real or complex indices l and m of a real or complex number
  70. legendreQ( l, x ) — Legendre function of the second kind of real or complex index l of a real or complex number
  71. legendreQ( l, m, x ) — associated Legendre function of the second kind of real or complex indices l and m of a real or complex number
  72. sphericalHarmonic( l, m, θ, φ ) — spherical harmonic of integer indices l and m and real numbers. Returns a complex number even if the result is purely real.
  73. chebyshevT( n, x ) — Chebyshev polynomial of the first kind of real or complex index n of a real or complex number
  74. chebyshevU( n, x ) — Chebyshev polynomial of the second kind of real or complex index n of a real or complex number

  75. Elliptic Integrals 椭圆积分

  76. ellipticF( x, m ) — incomplete elliptic integral of the first kind of a real or complex number with real or complex elliptic parameter m
  77. ellipticF( m ) — complete elliptic integral of the first kind of a real or complex elliptic parameter m
  78. ellipticK( m ) — complete elliptic integral of the first kind of a real or complex elliptic parameter m
  79. ellipticE( x, m ) — incomplete elliptic integral of the second kind of a real or complex number with real or complex elliptic parameter m
  80. ellipticE( m ) — complete elliptic integral of the second kind of a real or complex elliptic parameter m
  81. ellipticPi( n, x, m ) — incomplete elliptic integral of the third kind of a real or complex number with real or complex characteristic n and elliptic parameter m
  82. ellipticPi( n, m ) — complete elliptic integral of the third kind of a real or complex elliptic characteristic n and parameter m
  83. jacobiZeta( x, m ) — Jacobi zeta function of a real or complex number with real or complex elliptic parameter m, with the first argument of the same type as for elliptic integrals
  84. carlsonRC( x, y ) — degenerate Carlson symmetric elliptic integral of the first kind of real or complex numbers
  85. carlsonRD( x, y, z ) — degenerate Carlson symmetric elliptic integral of the third kind, or Carlson elliptic integral of the second kind, of real or complex numbers
  86. carlsonRF( x, y, z ) — Carlson symmetric elliptic integral of the first kind of real or complex numbers
  87. carlsonRG( x, y, z ) — Carlson completely symmetric elliptic integral of the second kind of real or complex numbers
  88. carlsonRJ( x, y, z, w ) — Carlson symmetric elliptic integral of the third kind of real or complex numbers

  89. Elliptic Functions 椭圆函数

  90. jacobiTheta( n, x, q ) — Jacobi theta function n of a real or complex number with real or complex nome q
  91. ellipticNome( m ) — elliptic nome q of a real or complex elliptic parameter m
  92. am( x, m ) — Jacobi amplitude of a real or complex number with real or complex elliptic parameter m
  93. sn( x, m ) — Jacobi elliptic sine of a real or complex number with real or complex elliptic parameter m
  94. cn( x, m ) — Jacobi elliptic cosine of a real or complex number with real or complex elliptic parameter m
  95. dn( x, m ) — Jacobi delta amplitude of a real or complex number with real or complex elliptic parameter m
  96. weierstrassRoots( g2, g3 ) — Weierstrass roots e1, e2 and e3 for real or complex invariants. Returned as an array.
  97. weierstrassHalfPeriods( g2, g3 ) — Weierstrass half periods w1 and w3 for real or complex invariants. Returned as an array. Consistent with evaluation of Weierstrass elliptic function in terms of Jacobi elliptic sine.
  98. weierstrassInvariants( w1, w3 ) — Weierstrass invariants g2 and g3 for real or complex half periods. Returned as an array.
  99. weierstrassP( x, g2, g3 ) — Weierstrass elliptic function ℘ of a real or complex number with real or complex invariants. Returned as a complex number for consistency.
  100. weierstrassPPrime( x, g2, g3 ) — derivative of the Weierstrass elliptic function ℘ of a real or complex number with real or complex invariants. Returned as a complex number for consistency.
  101. inverseWeierstrassP( x, g2, g3 ) — inverse Weierstrass elliptic function ℘ of a real or complex number with real or complex invariants. Returned as a complex number for consistency.
  102. kleinJ( x ) — Klein j-invariant of a complex number

  103. Hypergeometric Functions 超几何函数

  104. hypergeometric0F1( a, x ) — confluent hypergeometric function of a real or complex parameter a of a real or complex number
  105. hypergeometric1F1( a, b, x ) — confluent hypergeometric function of the first kind of real or complex parameters a and b of a real or complex number
  106. hypergeometricU( a, b, x ) — confluent hypergeometric function of the second kind of real or complex parameters a and b of a real or complex number
  107. whittakerM( k, m, x ) — Whittaker function of the first kind of real or complex parameters k and m of a real or complex number
  108. whittakerW( k, m, x ) — Whittaker function of the second kind of real or complex parameters k and m of a real or complex number
  109. hypergeometric2F1( a, b, c, x ) — Gauss hypergeometric function of real or complex parameters a, b and c of a real or complex number
  110. hypergeometric1F2( a, b, c, x ) — hypergeometric function of real or complex parameters a, b and c of a real or complex number
  111. hypergeometricPFQ( A, B, x ) — generalized hypergeometric function of arrays of real or complex parameters A and B of a real or complex number

  112. Gamma Functions 伽马函数

  113. beta( x, y ) — beta function of real or complex numbers
  114. beta( x, y, z ) — incomplete beta function Bx(y,z) of real or complex numbers, where x = 1 replicates the beta function
  115. beta( x, y, z, w ) — generalized incomplete beta function By(z,w) − Bx(z,w) of real or complex numbers
  116. betaRegularized( x, y, z ) — regularized incomplete beta function Ix(y,z) of real or complex numbers
  117. betaRegularized( x, y, z, w ) — generalized regularized incomplete beta function Iy(z,w) − Ix(z,w) of real or complex numbers

  118. factorial( n ) — factorial of a real or complex number
  119. factorial2( n ) — double factorial of a real or complex number
  120. binomial( n, m ) — binomial coefficient of real or complex numbers
  121. logGamma( x ) — logarithm of the gamma function of a real or complex number
  122. gamma( x ) — gamma function of a real or complex number
  123. gamma( x, y ) — upper incomplete gamma function Γ(x,y) of real or complex numbers
  124. gamma( x, 0, y ) — lower incomplete gamma function γ(x,y) of real or complex numbers
  125. gamma( x, y, z ) — generalized incomplete gamma function γ(x,z) − γ(x,y) of real or complex numbers
  126. gammaRegularized( x, y ) — regularized upper incomplete gamma function Q(x,y) of real or complex numbers
  127. gammaRegularized( x, y, z ) — generalized regularized incomplete gamma function Q(x,z) − Q(x,y) of real or complex numbers

  128. psi(x)=polygamma(x)=digamma( x ) — digamma function of a real or complex number
  129. psi(1,x)=polygamma(1,x) — polygamma function of a real or complex number

  130. Gamma-Type Functions

  131. erf( x ) — error function of a real or complex number
  132. erfc( x ) — complementary error function of a real or complex number
  133. erfi( x ) — imaginary error function of a real or complex number
  134. fresnelS( x ) — Fresnel sine integral of a real or complex number
  135. fresnelC( x ) — Fresnel cosine integral of a real or complex number
  136. Ei(x)=expIntegral( x ) — exponential integral of a real or complex number
  137. li(x)=logIntegral( x ) — logarithmic integral of a real or complex number
  138. si(x)=sinIntegral( x ) — sine integral of a real or complex number
  139. ci(x)=cosIntegral( x ) — cosine integral of a real or complex number
  140. shi(x)=sinhIntegral( x ) — hyperbolic sine integral of a real or complex number
  141. chi(x)=coshIntegral( x ) — hyperbolic cosine integral of a real or complex number
  142. En(n,x)=expIntegralE( n, x ) — generalized exponential integral of a real or complex order n of a real or complex number

  143. Zeta Functions

  144. zeta( x ) — Riemann zeta of a real or complex number
  145. eta(x)=dirichletEta( x ) — Dirichlet eta of a real or complex number
  146. bernoulli( n ) — Bernoulli number for index n
  147. harmonic( n ) — harmonic number for index n
  148. hurwitzZeta( x, a ) — Hurwitz zeta function of a real or complex number with real or complex parameter a

  149. Miscellaneous Functions

  150. chop( x ) — set real and complex parts smaller than 10−10 to zero
  151. chop( x, tolerance ) — set real and complex parts smaller than tolerance to zero
  152. kronecker( i, j ) — Kronecker delta δij for integer arguments
  153. piecewise( [ function, [begin,end] ], … ) — piecewise expression defined on an arbitrary number of subdomains returned as a function

Examples 复变函数例题

Reference

  • math handbook: chapter 10 complex function 复变函数
  • complex math:
  • 复变函数(史济怀)
  • 复变函数与积分变换(第二版)华中科大
  • 复变函数与积分变换
  • 复变函数同步辅导及习题全解-第四版-华东师大
  • 复变函数引论-下册(普里瓦洛夫)
  • 复变函数-西安交大第4版
  • 复变函数论例题选讲 
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