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# Complex Function复变函数图

## Function Search

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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特殊函数图

### 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(z,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. polylog( n,b,x ) — polylogarithm function of real or complex order n 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

## 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 ﻿