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# Complex Function复变函数图

## 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特殊函数图

### 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

## 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 ﻿