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, !n = n! × (-1)^n / e
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. complex2D
   1. re2D(x) and im2D(x) show for complex 2 curves of real and imag parts in real and imag domain.
   2. complex coloring
   3. color WebXR surface of complex function on complex plane
   4. complex animate(z) or complex2D(z) for phase animation in complex plane, the independent variable must be z.
   5. complex plot(z) for phase and/or modulus in complex plane, the independent variable must be z.
   6. plot complex(z) for phase and/or modulus in complex plane, the independent variable must be z.
2. complex3D
   1. complex function
   2. inverse complex function
   3. Riemann surface
   4. Complex Branches
   5. complex3D(x) for 3 dimensional graph, the independent variable must be x.

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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