Complex Function 复变函数图

List of Math function and operations 函数目录

Complex function in different domain or plane

Real domain

Complex domain

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) = log_b(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

inverse function

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



Hyperbolic Functions 双曲函数

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

inverse function

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



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



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



Orthogonal Polynomials 正交多项式

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



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



Elliptic Functions 椭圆函数

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



Hypergeometric Functions 超几何函数

112. hypergeometric0F1( a, x ) — confluent hypergeometric function of a real or complex parameter a of a real or complex number
113. 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
114. 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
115. whittakerM( k, m, x ) — Whittaker function of the first kind of real or complex parameters k and m of a real or complex number
116. whittakerW( k, m, x ) — Whittaker function of the second kind of real or complex parameters k and m of a real or complex number
117. hypergeometric2F1( a, b, c, x ) — Gauss hypergeometric function of real or complex parameters a, b and c of a real or complex number
118. hypergeometric1F2( a, b, c, x ) — hypergeometric function of real or complex parameters a, b and c of a real or complex number
119. hypergeometricPFQ( A, B, x ) — generalized hypergeometric function of arrays of real or complex parameters A and B of a real or complex number



Gamma Functions 伽马函数

120. beta( x, y ) — beta function of real or complex numbers
121. beta( x, y, z ) — incomplete beta function B_x(y,z) of real or complex numbers, where x = 1 replicates the beta function
122. beta( x, y, z, w ) — generalized incomplete beta function B_y(z,w) − B_x(z,w) of real or complex numbers
123. betaRegularized( x, y, z ) — regularized incomplete beta function I_x(y,z) of real or complex numbers
124. betaRegularized( x, y, z, w ) — generalized regularized incomplete beta function I_y(z,w) − I_x(z,w) of real or complex numbers

125. factorial( n ) — factorial of a real or complex number
126. factorial2( n ) — double factorial of a real or complex number
127. subfactorial( n ) — subfactorial of a real or complex number
128. pochhammer( x, n ) = risingfactorial(x,n) — Pochhammer symbol of real or complex numbers
129. binomial( n, m ) — binomial coefficient of real or complex numbers
130. multinomial( n1, n2, … ) — multinomial 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



Gamma-Type Functions

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

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. riemannXi( x ) — Riemman xi function of a real or complex number
158. bernoulli( n ) — Bernoulli number for index n
159. bernoulli( n,x ) — Bernoulli polynomial for index n of a real or complex number
160. H(x) = harmonic( n ) — harmonic number for index n
161. harmonic( n,x ) — harmonic number for index n from 1 to x
162. harmonic( n,a,x ) — harmonic number for index n from a to x
163. polylog( n,x ) — polylogarithm function of real or complex order n of a real or complex number
164. polylog( n,b,x ) — polylogarithm function of real or complex order n of a real or complex number
     

     Miscellaneous Functions

165. chop( x ) — set real and complex parts smaller than 10⁻¹⁰ to zero
166. chop( x, tolerance ) — set real and complex parts smaller than tolerance to zero
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. random( ) — random real number between zero and one
175. random( x ) — random real or complex number between zero and x
176. random( x, y ) — random real or complex number between x and y
177. kronecker( i, j ) — Kronecker delta δ_ij for real or complex arguments
178. kronecker( i, j, k, … ) — Kronecker delta δ_ijk… for an arbitrary number of real or complex arguments
179. 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) or complex2D(x) for phase animation in complex plane, the independent variable must be z.
3. complex plot(z) for phase and/or modulus in complex plane, the independent variable must be z.
4. plot complex(z) for phase and/or modulus in complex plane, the independent variable must be z.
5. complex2D show re2D(x) and im2D(x) for complex 2 curves of real and imag parts in real and imag domain, 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

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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