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Complex Function 复变函数

Content

  1. List of Functions 函数目录
  2. Complex function in different domain or plane
  3. Basic Function 初等复变函数
  4. circular function
  5. Special Function 特殊函数
  6. gamma Functions 伽马函数
  7. zeta Functions
  8. Reference

Function Search

  • Search function with ? in the input box, e.g.

  • search function in wiki. e.g.
  • serach function in Digital Library of Mathematical Functions NIST, e.g.
    erf

    List of Math function and operations 函数目录

    function reference

    Complex function in different domain or plane

  • plot with complex2D( ) in real domain for plane curve 2D
  • WebXR Surface 2D in complex domain and plane
  • plot with complex3D( ) in 3D space on complex plane
    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.

    Notice that Microsoft Internet Explorer IE did not support svg, so IE cannot show these graph, please use other browrer.

    Basic Function 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( x ,base) = 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

      math handbook chapter 12 special function

      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

      Polynomial function
    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, Dawson plus, it is the particular solution to the differential equation y'+2x*y=1
    156. Dawsonm(x) = erf(x)*exp(x*x)*sqrt(pi)/2, Dawson minus, it is the particular solution to the differential equation y'-2x*y=1

      Zeta Functions

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

      Miscellaneous Functions

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

    Reference

  • complex
  • complex math
  • math handbook: chapter 10 complex function 复变函数
  • math handbook: chapter 12 special function 特殊函数
  • 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