﻿﻿ Fractional Calculus Computer Algebra System, Electrochemistry Software, Massage + + =    ﻿

# Complex Function复变函数图

## Function Search

Input function name for search.

## 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.

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

## 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. pow( x, y ) — power of a real or complex number to a real or complex exponent
5. root( x, y ) — root of a real or complex number with real or complex degree
6. sqrt( x ) — square root of a real or complex number
7. cbrt( x ) — cubic root of a real or complex number

8. ### Logarithmic Functions 对数函数

9. exp( x ) — exponential of a real or complex number
10. W(x) = lambertW( x ) — principal branch of the Lambert W-function of a real number
11. W(k,x) = lambertW( k, x ) — real branches of the Lambert W-function of a real number for k = −1 or k = 0
12. #### inverse function

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 ) — natural logarithm of a real or complex number
16. log( x, base ) — logarithm of a real or complex number to a real or complex base
17. log10( x ) — the 10-base logarithm of a real or complex number
18. inverseW(x) = inverseLambertW( x ) — inverse of the Lambert W-function of a real number, = exp(x)*x

19. ### Circular Functions 三角函数

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

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

35. ### Hyperbolic Functions 双曲函数

36. sinh( x ) — hyperbolic sine of a real or complex number
37. cosh( x ) — hyperbolic cosine of a real or complex number
38. tanh( x ) — hyperbolic tangent of a real or complex number
39. coth( x ) — hyperbolic cotangent of a real or complex number
40. sech( x ) — hyperbolic secant of a real or complex number
41. csch( x ) — hyperbolic cosecant of a real or complex number
42. gudermannian( x ) — Gudermannian function of a real or complex number, = arctan( sinh(x) )
43. #### inverse function

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

51. ## Special Function特殊函数图

### Bessel Functions 贝塞耳函数

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

62. ### Bessel-Type Functions

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

73. ### Orthogonal Polynomials 正交多项式

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

84. ### Elliptic Integrals 椭圆积分

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

98. ### Elliptic Functions 椭圆函数

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

113. ### Hypergeometric Functions 超几何函数

114. hypergeometric0F1( a, x ) — confluent hypergeometric function of a real or complex parameter a of a real or complex number
115. 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
116. 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
117. whittakerM( k, m, x ) — Whittaker function of the first kind of real or complex parameters k and m of a real or complex number
118. whittakerW( k, m, x ) — Whittaker function of the second kind of real or complex parameters k and m of a real or complex number
119. hypergeometric2F1( a, b, c, x ) — Gauss hypergeometric function of real or complex parameters a, b and c of a real or complex number
120. hypergeometric1F2( a, b, c, x ) — hypergeometric function of real or complex parameters a, b and c of a real or complex number
121. hypergeometricPFQ( A, B, x ) — generalized hypergeometric function of arrays of real or complex parameters A and B of a real or complex number

122. ### Gamma Functions 伽马函数

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

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

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

140. ### Gamma-Type Functions

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

### Zeta Functions

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

### Miscellaneous Functions

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