AI Fractional Calculus Computer Algebra System, Electrochemistry Software, chat GPT, AI math GPT DrHuang.com | list | science | math | physics | chemistry | computer | medicine | help | ? | 中文
+ + + =

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

    Real domain

  • plot with complex2D( x ) in real domain for 2 curves of real and imag parts

    Complex domain

  • WebXR Surface 2D in complex domain and plane
  • complexplot( z ) in complex domain and plane
  • plot with complex3D( x ) 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.

    Real Function 实函数

    1. abs(x,y) = hypo(x, y) = sqrt(x*x+y*y) — absolute value of real number
    2. surd( x, n ) — real-valued root of a real number, n must be integer
    3. nthRoot(x,n) — real-valued root of a real number

      Basic Function 初等复变函数

      Basic Functions 基本初等函数

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

      Logarithmic Functions 对数函数

    13. ln(x) = log( x ) — natural logarithm of a real or complex number, inverse of exp(x)
    14. ln(n,x) = ln(n)(x) — the nth derivative of ln(x)
    15. log( x ) = ln(x) — natural logarithm of a real or complex number
    16. log( x ,base) = logb(x) — logarithm of a real or complex number to a real or complex base
    17. log10( x ) = log10(x) — the 10-base logarithm of a real or complex number
    18. W(x) = lambertW( x ) — principal branch of the Lambert W-function of a real number or complex number
    19. W(n,x) = lambertW( k, x ) — branch of integer index k of the Lambert W function of a real or complex number
    20. doubleLambert( x, y ) — principle branch of a double Lambert function of two real or complex numbers
    21. doubleLambert( n, x, y ) — arbitrary branch of integral index n of a double Lambert function of two real or complex numbers
    22. logisticSigmoid( x ) — logistic sigmoid of a real or complex number
    23. wrightOmega( x ) — Wright omega function of a real or complex number

      Circular Functions 三角函数

    24. sin( x ) — sine of a real or complex number
    25. cos( x ) — cosine of a real or complex number
    26. tan( x ) — tangent of a real or complex number
    27. cot( x ) — cotangent of a real or complex number
    28. sec( x ) — secant of a real or complex number
    29. csc( x ) — cosecant of a real or complex number
    30. inverse function

    31. asin(x) = arcsin( x ) — inverse sine of a real or complex number
    32. acos(x) = arccos( x ) — inverse cosine of a real or complex number
    33. atan(x) = arctan( x ) — inverse tangent of a real or complex number
    34. acot(x) = arccot( x ) — inverse cotangent of a real or complex number
    35. asec(x) = arcsec( x ) — inverse secant of a real or complex number
    36. acsc(x) = arccsc( x ) — inverse cosecant of a real or complex number
    37. atan2(y,x) — inverse tangent of real number

    38. Hyperbolic Functions 双曲函数

    39. sinh( x ) — hyperbolic sine of a real or complex number
    40. cosh( x ) — hyperbolic cosine of a real or complex number
    41. tanh( x ) — hyperbolic tangent of a real or complex number
    42. coth( x ) — hyperbolic cotangent of a real or complex number
    43. sech( x ) — hyperbolic secant of a real or complex number
    44. csch( x ) — hyperbolic cosecant of a real or complex number
    45. inverse function

    46. asinh(x) = arcsinh( x ) — inverse hyperbolic sine of a real or complex number
    47. acosh(x) = arccosh( x ) — inverse hyperbolic cosine of a real or complex number
    48. atanh(x) = arctanh( x ) — inverse hyperbolic tangent of a real or complex number
    49. acoth(x) = arccoth( x ) — inverse hyperbolic cotangent of a real or complex number
    50. asech(x) = arcsech( x ) — inverse secant of a real or complex number
    51. acsch(x) = arccsch( x ) — inverse hyperbolic cosecant of a real or complex number

    52. Trigonometric Functions

    53. sinc( x ) = sin(x)/x — cardinal sine of a real or complex number
    54. sinc(x,y) = sinc( abs(x,y) )
    55. gudermannian( x ) = arctan( sinh(x) ) — Gudermannian function of a real or complex number,
    56. haversine( x ) = sin(x/2)^2 -— haversine of a real or complex number

      inverse function

    57. inverseGudermannian( x ) = arctanh( sin(x) ) — inverse Gudermannian function of a real or complex number,
    58. inverseHaversine( x ) = inverse( haversine(x) ) = 2asin(sqrt(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

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

    69. Bessel-Type Functions

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

    80. Orthogonal Polynomials 正交多项式

      Polynomial function
    81. hermite( n, x ) — Hermite polynomial of real or complex index n of a real or complex number
    82. laguerre( n, x ) — Laguerre polynomial of real or complex index n of a real or complex number
    83. 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
    84. legendreP( l, x ) — Legendre polynomial of real or complex index l of a real or complex number
    85. legendreP( l, m, x ) — associated Legendre polynomial of real or complex indices l and m of a real or complex number
    86. legendreQ( l, x ) — Legendre function of the second kind of real or complex index l of a real or complex number
    87. 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
    88. chebyshevT( n, x ) — Chebyshev polynomial of the first kind of real or complex index n of a real or complex number
    89. chebyshevU( n, x ) — Chebyshev polynomial of the second kind of real or complex index n of a real or complex number
    90. 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.

    91. Elliptic Integrals 椭圆积分

    92. ellipticF( x, m ) — incomplete elliptic integral of the first kind of a real or complex number with real or complex elliptic parameter m
    93. ellipticF( m ) — complete elliptic integral of the first kind of a real or complex elliptic parameter m
    94. ellipticK( m ) — complete elliptic integral of the first kind of a real or complex elliptic parameter m
    95. ellipticE( x, m ) — incomplete elliptic integral of the second kind of a real or complex number with real or complex elliptic parameter m
    96. ellipticE( m ) — complete elliptic integral of the second kind of a real or complex elliptic parameter m
    97. 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
    98. ellipticPi( n, m ) — complete elliptic integral of the third kind of a real or complex elliptic characteristic n and parameter m
    99. 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
    100. carlsonRC( x, y ) — degenerate Carlson symmetric elliptic integral of the first kind of real or complex numbers
    101. 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
    102. carlsonRF( x, y, z ) — Carlson symmetric elliptic integral of the first kind of real or complex numbers
    103. carlsonRG( x, y, z ) — Carlson completely symmetric elliptic integral of the second kind of real or complex numbers
    104. carlsonRJ( x, y, z, w ) — Carlson symmetric elliptic integral of the third kind of real or complex numbers

    105. Elliptic Functions 椭圆函数

    106. jacobiTheta( n, x, q ) — Jacobi theta function n of a real or complex number with real or complex nome q
    107. ellipticNome( m ) — elliptic nome q of a real or complex elliptic parameter m
    108. am( x, m ) — Jacobi amplitude of a real or complex number with real or complex elliptic parameter m
    109. sn( x, m ) — Jacobi elliptic sine of a real or complex number with real or complex elliptic parameter m
    110. cn( x, m ) — Jacobi elliptic cosine of a real or complex number with real or complex elliptic parameter m
    111. dn( x, m ) — Jacobi delta amplitude of a real or complex number with real or complex elliptic parameter m
    112. weierstrass(x)
    113. weierstrassRoots( g2, g3 ) — Weierstrass roots e1, e2 and e3 for real or complex invariants. Returned as an array.
    114. 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.
    115. weierstrassInvariants( w1, w3 ) — Weierstrass invariants g2 and g3 for real or complex half periods. Returned as an array.
    116. 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.
    117. 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.
    118. 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.
    119. kleinJ( x ) — Klein j-invariant of a complex number

    120. Hypergeometric Functions 超几何函数

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

    129. Gamma Functions 伽马函数

    130. beta( x, y ) — beta function of real or complex numbers
    131. beta( x, y, z ) — incomplete beta function Bx(y,z) of real or complex numbers, where x = 1 replicates the beta function
    132. beta( x, y, z, w ) — generalized incomplete beta function By(z,w) − Bx(z,w) of real or complex numbers
    133. betaRegularized( x, y, z ) — regularized incomplete beta function Ix(y,z) of real or complex numbers
    134. betaRegularized( x, y, z, w ) — generalized regularized incomplete beta function Iy(z,w) − Ix(z,w) of real or complex numbers

    135. factorial( n ) — factorial of a real or complex number
    136. factorial2( n ) — double factorial of a real or complex number
    137. subfactorial( n ) — subfactorial of a real or complex number
    138. pochhammer( x, n ) = risingfactorial(x,n) — Pochhammer symbol of real or complex numbers
    139. binomial( n, m ) — binomial coefficient of real or complex numbers
    140. multinomial( n1, n2, … ) — multinomial coefficient of real or complex numbers

    141. gamma( x ) — gamma function of a real or complex number
    142. gamma( x, y ) — upper incomplete gamma function Γ(x,y) of real or complex numbers
    143. gamma( x, 0, y ) — lower incomplete gamma function γ(x,y) of real or complex numbers
    144. gamma( x, y, z ) — generalized incomplete gamma function γ(x,z) − γ(x,y) of real or complex numbers
    145. GammaQ(x, y) = gammaRegularized( x, y ) — regularized upper incomplete gamma function Q(x,y) of real or complex numbers
    146. GammaQ(x,y,z) = gammaRegularized( x, y, z ) — generalized regularized incomplete gamma function Q(x,z) − Q(x,y) of real or complex numbers

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

    150. Gamma-Type Functions

    151. erf( x ) — error function of a real or complex number
    152. erfc( x ) = 1-erf(x) — complementary error function of a real or complex number,
    153. erfi( x ) — imaginary error function of a real or complex number
    154. fresnelS( x ) — Fresnel sine integral of a real or complex number
    155. fresnelC( x ) — Fresnel cosine integral of a real or complex number
    156. Ei(x) = expIntegral( x ) — exponential integral of a real or complex number
    157. En(n,x) = expIntegralE( n, x ) — generalized exponential integral of a real or complex order n of a real or complex number
    158. li(x) = logIntegral( x ) — logarithmic integral of a real or complex number
    159. si(x) = sinIntegral( x ) — sine integral of a real or complex number
    160. ci(x) = cosIntegral( x ) — cosine integral of a real or complex number
    161. shi(x) = sinhIntegral( x ) — hyperbolic sine integral of a real or complex number
    162. chi(x) = coshIntegral( x ) — hyperbolic cosine integral of a real or complex number
    163. 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
    164. 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

    165. zeta( x ) — Riemann zeta of a real or complex number
    166. zeta(z,a) = hurwitzZeta( x, a ) — Hurwitz zeta function of a real or complex number with real or complex parameter a
    167. eta(x) = dirichletEta( x ) — Dirichlet eta of a real or complex number
    168. bernoulli( n ) — Bernoulli number for index n
    169. bernoulli( n,x ) — Bernoulli polynomial for index n of a real or complex number
    170. H(x) = harmonic( n ) — harmonic number for index n
    171. harmonic( n,x ) — harmonic number for index n from 1 to x
    172. harmonic( n,a,x ) — harmonic number for index n from a to x
    173. polylog( n,x ) — polylogarithm function of real or complex order n of a real or complex number
    174. polylog( n,b,x ) — polylogarithm function of real or complex order n of a real or complex number

      Miscellaneous Functions

    175. chop( x ) — set real and complex parts smaller than 10−10 to zero
    176. chop( x, tolerance ) — set real and complex parts smaller than tolerance to zero
    177. round( x ) — closest integer to a real or complex number
    178. round( x, y ) — closest integer multiple of y to a real or complex number
    179. ceiling( x ) — closest integer greater than a real or complex number
    180. floor( x ) — closest integer less than a real or complex number
    181. sgn(x) = sign( x ) — signum function of a real or complex number
    182. integerPart( x ) — integer part of a real or complex number
    183. fractionalPart( x ) — fractional part of a real or complex number
    184. random( ) — random real number between zero and one
    185. random( x ) — random real or complex number between zero and x
    186. random( x, y ) — random real or complex number between x and y
    187. kronecker( i, j ) — Kronecker delta δij for real or complex arguments
    188. kronecker( i, j, k, … ) — Kronecker delta δijk… for an arbitrary number of real or complex arguments
    189. piecewise( [ function, [begin,end] ], … ) — piecewise expression defined on an arbitrary number of subdomains returned as a function

    Complex

    1. complex - complex function - complex math
    2. math handbook chapter 10 complex function
    3. complex animate(z) for phase animation, the independent variable must be z.
    4. complex plot(z) for phase and/or modulus, the independent variable must be z.
    5. complex2D(x) for complex 2 curves of real and imag parts, the independent variable must be x.
    6. complex3D(x) for 3 dimensional graph, the independent variable must be x.
    7. color WebXR surface of complex function on complex plane
    8. Riemann surface - Complex Branches - complex coloring

    References

    1. Complex analysis
    
    See Also