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# Complex Function复变函数图

## 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特殊函数图

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

## References

1. math handbook content 2 chapter 10 complex function
2. math handbook content 3 chapter 10 complex function
3. math handbook content 4 chapter 10 complex function
4. Complex analysis
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See Also