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Complex Function 复变函数

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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. abs(x, y) = hypo(x, y) = sqrt(x*x+y*y) — absolute value of real number
  3. arg( x ) — argument of a real or complex number
  4. arg( x, y ) = arg(complex(x,y)) = atant2(y,x) — argument of a real or complex number
  5. pow( x, y ) — power of a real or complex number to a real or complex exponent
  6. root( x, y ) — root of a real or complex number with real or complex degree
  7. sqrt( x ) — square root of a real or complex number
  8. cbrt( x ) — cubic root of a real or complex number
  9. exp( x ) — exponential of a real or complex number
  10. exp(x)*x = inverseW(x) = inverseLambertW( x ) — inverse of the Lambert W-function of a real number,or complex number
  11. nthRoot(x,n) = surd( x, n ) — real-valued root of a real number

    Logarithmic Functions 对数函数

  12. ln(x) = log( x ) — natural logarithm of a real or complex number, inverse of exp(x)
  13. ln(n,x) = ln(n)(x) — the nth derivative of ln(x)
  14. log( x ) = ln(x) — natural logarithm of a real or complex number
  15. log(base, x ) = logb(x) — logarithm of a real or complex number to a real or complex base
  16. log10( x ) = log10(x) — the 10-base logarithm of a real or complex number
  17. W(x) = lambertW( x ) — principal branch of the Lambert W-function of a real number or complex number
  18. W(k,x) = lambertW( k, x ) — branch of integer index k of the Lambert W function of a real or complex number

    Circular Functions 三角函数

  19. sin( x ) — sine of a real or complex number
  20. cos( x ) — cosine of a real or complex number
  21. tan( x ) — tangent of a real or complex number
  22. cot( x ) — cotangent of a real or complex number
  23. sec( x ) — secant of a real or complex number
  24. csc( x ) — cosecant of a real or complex number
  25. inverse function

  26. asin(x) = arcsin( x ) — inverse sine of a real or complex number
  27. acos(x) = arccos( x ) — inverse cosine of a real or complex number
  28. atan(x) = arctan( x ) — inverse tangent of a real or complex number
  29. acot(x) = arccot( x ) — inverse cotangent of a real or complex number
  30. asec(x) = arcsec( x ) — inverse secant of a real or complex number
  31. acsc(x) = arccsc( x ) — inverse cosecant of a real or complex number
  32. atan2(y,x) — inverse tangent of real number

  33. Hyperbolic Functions 双曲函数

  34. sinh( x ) — hyperbolic sine of a real or complex number
  35. cosh( x ) — hyperbolic cosine of a real or complex number
  36. tanh( x ) — hyperbolic tangent of a real or complex number
  37. coth( x ) — hyperbolic cotangent of a real or complex number
  38. sech( x ) — hyperbolic secant of a real or complex number
  39. csch( x ) — hyperbolic cosecant of a real or complex number
  40. inverse function

  41. asinh(x) = arcsinh( x ) — inverse hyperbolic sine of a real or complex number
  42. acosh(x) = arccosh( x ) — inverse hyperbolic cosine of a real or complex number
  43. atanh(x) = arctanh( x ) — inverse hyperbolic tangent of a real or complex number
  44. acoth(x) = arccoth( x ) — inverse hyperbolic cotangent of a real or complex number
  45. asech(x) = arcsech( x ) — inverse secant of a real or complex number
  46. acsch(x) = arccsch( x ) — inverse hyperbolic cosecant of a real or complex number

  47. Trigonometric Functions

  48. sinc( x ) — cardinal sine of a real or complex number
  49. sinc(x,y) = sinc(abs(x,y))
  50. gd(x) = gudermannian( x ) — Gudermannian function of a real or complex number, = arctan( sinh(x) )
  51. inverseGudermannian( x ) — inverse Gudermannian function of a real or complex number, = arctanh( sin(x) )
  52. haversine( x ) — haversine of a real or complex number
  53. inverseHaversine( x ) — inverse haversine of a real or complex number

    Special Function 特殊函数

    Bessel Functions 贝塞耳函数

  54. besselJ( n, x ) — Bessel function of the first kind of real or complex order n of a real or complex number
  55. besselJZero( n, m )mth zero of the Bessel function of the first kind of positive order n
  56. besselJZero( n, m, true )mth zero of the first derivative of the Bessel function of the first kind of positive order n
  57. besselY( n, x ) — Bessel function of the second kind of real or complex order n of a real or complex number
  58. besselYZero( n, m )mth zero of the Bessel function of the second kind of positive order n
  59. besselYZero( n, m, true )mth zero of the first derivative of the Bessel function of the second kind of positive order n
  60. besselI( n, x ) — modified Bessel function of the first kind of real or complex order n of a real or complex number
  61. besselK( n, x ) — modified Bessel function of the second kind of real or complex order n of a real or complex number
  62. hankel1( n, x ) — Hankel function of the first kind of real or complex order n of a real or complex number
  63. hankel2( n, x ) — Hankel function of the second kind of real or complex order n of a real or complex number

  64. Bessel-Type Functions

  65. Ai(x) = airyAi( x ) — Airy function of the first kind of a real or complex number
  66. AiPrime(x) = airyAiPrime( x ) — derivative of the Airy function of the first kind of a real or complex number
  67. Bi(x) = airyBi( x ) — Airy function of the second kind of a real or complex number
  68. BiPrime(x) = airyBiPrime( x ) — derivative of the Airy function of the second kind of a real or complex number
  69. sphericalBesselJ( n, x ) — spherical Bessel function of the first kind of real or complex order n of a real or complex number
  70. sphericalBesselY( n, x ) — spherical Bessel function of the second kind of real or complex order n of a real or complex number
  71. sphericalHankel1( n, x ) — spherical Hankel function of the first kind of real or complex order n of a real or complex number
  72. sphericalHankel2( n, x ) — spherical Hankel function of the second kind of real or complex order n of a real or complex number
  73. struveH( n, x ) — Struve function of real or complex order n of a real or complex number
  74. struveL( n, x ) — modified Struve function of real or complex order n of a real or complex number

  75. Orthogonal Polynomials 正交多项式

  76. hermite( n, x ) — Hermite polynomial of real or complex index n of a real or complex number
  77. laguerre( n, x ) — Laguerre polynomial of real or complex index n of a real or complex number
  78. 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
  79. legendreP( l, x ) — Legendre polynomial of real or complex index l of a real or complex number
  80. legendreP( l, m, x ) — associated Legendre polynomial of real or complex indices l and m of a real or complex number
  81. legendreQ( l, x ) — Legendre function of the second kind of real or complex index l of a real or complex number
  82. 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
  83. chebyshevT( n, x ) — Chebyshev polynomial of the first kind of real or complex index n of a real or complex number
  84. chebyshevU( n, x ) — Chebyshev polynomial of the second kind of real or complex index n of a real or complex number
  85. 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.

  86. Elliptic Integrals 椭圆积分

  87. ellipticF( x, m ) — incomplete elliptic integral of the first kind of a real or complex number with real or complex elliptic parameter m
  88. ellipticF( m ) — complete elliptic integral of the first kind of a real or complex elliptic parameter m
  89. ellipticK( m ) — complete elliptic integral of the first kind of a real or complex elliptic parameter m
  90. ellipticE( x, m ) — incomplete elliptic integral of the second kind of a real or complex number with real or complex elliptic parameter m
  91. ellipticE( m ) — complete elliptic integral of the second kind of a real or complex elliptic parameter m
  92. 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
  93. ellipticPi( n, m ) — complete elliptic integral of the third kind of a real or complex elliptic characteristic n and parameter m
  94. 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
  95. carlsonRC( x, y ) — degenerate Carlson symmetric elliptic integral of the first kind of real or complex numbers
  96. 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
  97. carlsonRF( x, y, z ) — Carlson symmetric elliptic integral of the first kind of real or complex numbers
  98. carlsonRG( x, y, z ) — Carlson completely symmetric elliptic integral of the second kind of real or complex numbers
  99. carlsonRJ( x, y, z, w ) — Carlson symmetric elliptic integral of the third kind of real or complex numbers

  100. Elliptic Functions 椭圆函数

  101. jacobiTheta( n, x, q ) — Jacobi theta function n of a real or complex number with real or complex nome q
  102. ellipticNome( m ) — elliptic nome q of a real or complex elliptic parameter m
  103. am( x, m ) — Jacobi amplitude of a real or complex number with real or complex elliptic parameter m
  104. sn( x, m ) — Jacobi elliptic sine of a real or complex number with real or complex elliptic parameter m
  105. cn( x, m ) — Jacobi elliptic cosine of a real or complex number with real or complex elliptic parameter m
  106. dn( x, m ) — Jacobi delta amplitude of a real or complex number with real or complex elliptic parameter m
  107. weierstrass(x)
  108. weierstrassRoots( g2, g3 ) — Weierstrass roots e1, e2 and e3 for real or complex invariants. Returned as an array.
  109. 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.
  110. weierstrassInvariants( w1, w3 ) — Weierstrass invariants g2 and g3 for real or complex half periods. Returned as an array.
  111. 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.
  112. 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.
  113. 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.
  114. kleinJ( x ) — Klein j-invariant of a complex number

  115. Hypergeometric Functions 超几何函数

  116. hypergeometric0F1( a, x ) — confluent hypergeometric function of a real or complex parameter a of a real or complex number
  117. 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
  118. 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
  119. whittakerM( k, m, x ) — Whittaker function of the first kind of real or complex parameters k and m of a real or complex number
  120. whittakerW( k, m, x ) — Whittaker function of the second kind of real or complex parameters k and m of a real or complex number
  121. hypergeometric2F1( a, b, c, x ) — Gauss hypergeometric function of real or complex parameters a, b and c of a real or complex number
  122. hypergeometric1F2( a, b, c, x ) — hypergeometric function of real or complex parameters a, b and c of a real or complex number
  123. hypergeometricPFQ( A, B, x ) — generalized hypergeometric function of arrays of real or complex parameters A and B of a real or complex number

  124. Gamma Functions 伽马函数

  125. beta( x, y ) — beta function of real or complex numbers
  126. beta( x, y, z ) — incomplete beta function Bx(y,z) of real or complex numbers, where x = 1 replicates the beta function
  127. beta( x, y, z, w ) — generalized incomplete beta function By(z,w) − Bx(z,w) of real or complex numbers
  128. betaRegularized( x, y, z ) — regularized incomplete beta function Ix(y,z) of real or complex numbers
  129. betaRegularized( x, y, z, w ) — generalized regularized incomplete beta function Iy(z,w) − Ix(z,w) of real or complex numbers

  130. factorial( n ) — factorial of a real or complex number
  131. factorial2( n ) — double factorial of a real or complex number
  132. binomial( n, m ) — binomial coefficient of real or complex numbers
  133. gamma( x ) — gamma function of a real or complex number
  134. gamma( x, y ) — upper incomplete gamma function Γ(x,y) of real or complex numbers
  135. gamma( x, 0, y ) — lower incomplete gamma function γ(x,y) of real or complex numbers
  136. gamma( x, y, z ) — generalized incomplete gamma function γ(x,z) − γ(x,y) of real or complex numbers
  137. gammaQ(x, y) = gammaRegularized( x, y ) — regularized upper incomplete gamma function Q(x,y) of real or complex numbers
  138. gammaQ(x,y,z) = gammaRegularized( x, y, z ) — generalized regularized incomplete gamma function Q(x,z) − Q(x,y) of real or complex numbers

  139. logGamma( x ) — logarithm of the gamma function of a real or complex number
  140. psi(x) = polygamma(x) = digamma( x ) = d/dx logGamma(x) — digamma function of a real or complex number
  141. psi(n,x) = polygamma(n,x) — polygamma function of positive integer order of a real or complex number

  142. Gamma-Type Functions

  143. erf( x ) — error function of a real or complex number
  144. erfc( x ) — complementary error function of a real or complex number, = 1-erf(x)
  145. erfi( x ) — imaginary error function of a real or complex number
  146. fresnelS( x ) — Fresnel sine integral of a real or complex number
  147. fresnelC( x ) — Fresnel cosine integral of a real or complex number
  148. Ei(x) = expIntegral( x ) — exponential integral of a real or complex number
  149. En(n,x) = expIntegralE( n, x ) — generalized exponential integral of a real or complex order n of a real or complex number
  150. li(x) = logIntegral( x ) — logarithmic integral of a real or complex number
  151. Si(x) = sinIntegral( x ) — sine integral of a real or complex number
  152. Ci(x) = cosIntegral( x ) — cosine integral of a real or complex number
  153. Shi(x) = sinhIntegral( x ) — hyperbolic sine integral of a real or complex number
  154. Chi(x) = coshIntegral( x ) — hyperbolic cosine integral of a real or complex number
  155. Dawson(x) = erfi(x)*exp(-x*x)*sqrt(pi)/2

    Zeta Functions

  156. zeta( x ) — Riemann zeta of a real or complex number
  157. zeta(x,a) = hurwitzZeta( x, a ) — Hurwitz zeta function of a real or complex number with real or complex parameter a
  158. eta(x) = dirichletEta( x ) — Dirichlet eta of a real or complex number
  159. bernoulli( n ) — Bernoulli number for index n
  160. bernoulli( n,x ) — Bernoulli polynomial for index n of a real or complex number
  161. H(x) = harmonic( n ) — harmonic number for index n
  162. harmonic( n,x ) — harmonic number for index n from 1 to x
  163. harmonic( n,a,x ) — harmonic number for index n from a to x
  164. polylog( n,x ) — polylogarithm function of real or complex order n of a real or complex number
  165. dilog( x ) = polylog(2,x) — polylogarithm function of a real or complex number

    Miscellaneous Functions

  166. chop( x ) — set real and complex parts smaller than 10−10 to zero
  167. chop( x, tolerance ) — set real and complex parts smaller than tolerance to zero
  168. piecewise( [ function, [begin,end] ], … ) — piecewise expression defined on an arbitrary number of subdomains returned as a function
  169. round( x ) — closest integer to a real or complex number
  170. round( x, y ) — closest integer multiple of y to a real or complex number
  171. ceiling( x ) — closest integer greater than a real or complex number
  172. floor( x ) — closest integer less than a real or complex number
  173. sgn(x) = sign( x ) — signum function of a real or complex number
  174. integerPart( x ) — integer part of a real or complex number
  175. fractionalPart( x ) — fractional part of a real or complex number
  176. kronecker( i, j ) — Kronecker delta δij for real or complex arguments
  177. kronecker( i, j, k, … ) — Kronecker delta δijk… for an arbitrary number of real or complex arguments

Curve graph 曲线图

Examples 例题

Reference

  • complex
  • complex math
  • math handbook: chapter 10 complex function 复变函数
  • Digital Library of Mathematical Functions
  • 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 
    See Also