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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

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. inverse function

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

34. Hyperbolic Functions 双曲函数

35. sinh( x ) — hyperbolic sine of a real or complex number
36. cosh( x ) — hyperbolic cosine of a real or complex number
37. tanh( x ) — hyperbolic tangent of a real or complex number
38. coth( x ) — hyperbolic cotangent of a real or complex number
39. sech( x ) — hyperbolic secant of a real or complex number
40. csch( x ) — hyperbolic cosecant of a real or complex number
41. inverse function

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

48. Trigonometric Functions

49. sinc( x ) = sin(x)/x — cardinal sine of a real or complex number
50. sinc(x,y) = sinc( abs(x,y) )
51. gudermannian( x ) = arctan( sinh(x) ) — Gudermannian function of a real or complex number,
52. haversine( x ) = sin(x/2)^2 -— haversine of a real or complex number

inverse function

53. inverseGudermannian( x ) = arctanh( sin(x) ) — inverse Gudermannian function of a real or complex number,
54. 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

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

65. Bessel-Type Functions

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

76. Orthogonal Polynomials 正交多项式

Polynomial function
77. hermite( n, x ) — Hermite polynomial of real or complex index n of a real or complex number
78. laguerre( n, x ) — Laguerre polynomial of real or complex index n of a real or complex number
79. 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
80. legendreP( l, x ) — Legendre polynomial of real or complex index l of a real or complex number
81. legendreP( l, m, x ) — associated Legendre polynomial of real or complex indices l and m of a real or complex number
82. legendreQ( l, x ) — Legendre function of the second kind of real or complex index l of a real or complex number
83. 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
84. chebyshevT( n, x ) — Chebyshev polynomial of the first kind of real or complex index n of a real or complex number
85. chebyshevU( n, x ) — Chebyshev polynomial of the second kind of real or complex index n of a real or complex number
86. 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.

87. Elliptic Integrals 椭圆积分

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

101. Elliptic Functions 椭圆函数

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

116. Hypergeometric Functions 超几何函数

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

125. Gamma Functions 伽马函数

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

131. factorial( n ) — factorial of a real or complex number
132. factorial2( n ) — double factorial of a real or complex number
133. binomial( n, m ) — binomial coefficient of real or complex numbers
134. multinomial( n1, n2, … ) — multinomial coefficient of real or complex numbers
135. pochhammer( x, n ) — Pochhammer symbol of real or complex numbers

136. gamma( x ) — gamma function of a real or complex number
137. gamma( x, y ) — upper incomplete gamma function Γ(x,y) of real or complex numbers
138. gamma( x, 0, y ) — lower incomplete gamma function γ(x,y) of real or complex numbers
139. gamma( x, y, z ) — generalized incomplete gamma function γ(x,z) − γ(x,y) of real or complex numbers
140. GammaQ(x, y) = gammaRegularized( x, y ) — regularized upper incomplete gamma function Q(x,y) of real or complex numbers
141. GammaQ(x,y,z) = gammaRegularized( x, y, z ) — generalized regularized incomplete gamma function Q(x,z) − Q(x,y) of real or complex numbers

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

145. Gamma-Type Functions

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

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

Miscellaneous Functions

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

References

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
9. 2D surface - 3D surface
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