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Complex Function 复变函数

Curve graph 曲线图

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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Basic Functions 基本初等函数

abs( x ) — absolute value of a real or complex number

arg( x ) — argument of a real or complex number

pow( x, y ) — power of a real or complex number to a real or complex exponent

root( x, y ) — root of a real or complex number with real or complex degree

sqrt( x ) — square root of a real or complex number

cbrt( x ) — cubic root of a real or complex number


Logarithmic Functions 对数函数

exp( x ) — exponential of a real or complex number

ln(x)=log( x ) — natural logarithm of a real or complex number

log( x, base ) — logarithm of a real or complex number to a real or complex base

lambertW( x ) — principal branch of the Lambert W-function of a real number

lambertW( k, x ) — real branches of the Lambert W-function of a real number for k = −1 or k = 0


Circular Functions 三角函数

sin( x ) — sine of a real or complex number

cos( x ) — cosine of a real or complex number

tan( x ) — tangent of a real or complex number

cot( x ) — cotangent of a real or complex number

sec( x ) — secant of a real or complex number

csc( x ) — cosecant of a real or complex number

arcsin( x ) — inverse sine of a real or complex number

arccos( x ) — inverse cosine of a real or complex number

arctan( x ) — inverse tangent of a real or complex number

arccot( x ) — inverse cotangent of a real or complex number

arcsec( x ) — inverse secant of a real or complex number

arccsc( x ) — inverse cosecant of a real or complex number

sinc( x ) — cardinal sine of a real or complex number


Hyperbolic Functions 双曲函数

sinh( x ) — hyperbolic sine of a real or complex number

cosh( x ) — hyperbolic cosine of a real or complex number

tanh( x ) — hyperbolic tangent of a real or complex number

coth( x ) — hyperbolic cotangent of a real or complex number

sech( x ) — hyperbolic secant of a real or complex number

csch( x ) — hyperbolic cosecant of a real or complex number

arcsinh( x ) — inverse hyperbolic sine of a real or complex number

arccosh( x ) — inverse hyperbolic cosine of a real or complex number

arctanh( x ) — inverse hyperbolic tangent of a real or complex number

arccoth( x ) — inverse hyperbolic cotangent of a real or complex number

arcsech( x ) — inverse secant of a real or complex number

arccsch( x ) — inverse hyperbolic cosecant of a real or complex number

gudermannian( x ) — Gudermannian function of a real or complex number

inverseGudermannian( x ) — inverse Gudermannian function of a real or complex number


Special Function 特殊函数

Examples 复变函数例题


Bessel Functions 贝塞耳函数

besselJ( n, x ) — Bessel function of the first kind of real or complex order n of a real or complex number

besselJZero( n, m )mth zero of the Bessel function of the first kind of positive order n

besselJZero( n, m, true )mth zero of the first derivative of the Bessel function of the first kind of positive order n

besselY( n, x ) — Bessel function of the second kind of real or complex order n of a real or complex number

besselYZero( n, m )mth zero of the Bessel function of the second kind of positive order n

besselYZero( n, m, true )mth zero of the first derivative of the Bessel function of the second kind of positive order n

besselI( n, x ) — modified Bessel function of the first kind of real or complex order n of a real or complex number

besselK( n, x ) — modified Bessel function of the second kind of real or complex order n of a real or complex number

hankel1( n, x ) — Hankel function of the first kind of real or complex order n of a real or complex number

hankel2( n, x ) — Hankel function of the second kind of real or complex order n of a real or complex number


Bessel-Type Functions

Ai(x)=airyAi( x ) — Airy function of the first kind of a real or complex number

airyAiPrime( x ) — derivative of the Airy function of the first kind of a real or complex number

Bi(x)=airyBi( x ) — Airy function of the second kind of a real or complex number

airyBiPrime( x ) — derivative of the Airy function of the second kind of a real or complex number

sphericalBesselJ( n, x ) — spherical Bessel function of the first kind of real or complex order n of a real or complex number

sphericalBesselY( n, x ) — spherical Bessel function of the second kind of real or complex order n of a real or complex number

sphericalHankel1( n, x ) — spherical Hankel function of the first kind of real or complex order n of a real or complex number

sphericalHankel2( n, x ) — spherical Hankel function of the second kind of real or complex order n of a real or complex number

struveH( n, x ) — Struve function of real or complex order n of a real or complex number

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

hermite( n, x ) — Hermite polynomial of real or complex index n of a real or complex number

laguerre( n, x ) — Laguerre polynomial of real or complex index n of a real or complex number

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

legendreP( l, x ) — Legendre polynomial of real or complex index l of a real or complex number

legendreP( l, m, x ) — associated Legendre polynomial of real or complex indices l and m of a real or complex number

legendreQ( l, x ) — Legendre function of the second kind of real or complex index l of a real or complex number

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

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.

chebyshevT( n, x ) — Chebyshev polynomial of the first kind of real or complex index n of a real or complex number

chebyshevU( n, x ) — Chebyshev polynomial of the second kind of real or complex index n of a real or complex number


Elliptic Integrals 椭圆积分

ellipticF( x, m ) — incomplete elliptic integral of the first kind of a real or complex number with real or complex elliptic parameter m

ellipticF( m ) — complete elliptic integral of the first kind of a real or complex elliptic parameter m

ellipticK( m ) — complete elliptic integral of the first kind of a real or complex elliptic parameter m

ellipticE( x, m ) — incomplete elliptic integral of the second kind of a real or complex number with real or complex elliptic parameter m

ellipticE( m ) — complete elliptic integral of the second kind of a real or complex elliptic parameter m

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

ellipticPi( n, m ) — complete elliptic integral of the third kind of a real or complex elliptic characteristic n and parameter m

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

carlsonRC( x, y ) — degenerate Carlson symmetric elliptic integral of the first kind of real or complex numbers

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

carlsonRF( x, y, z ) — Carlson symmetric elliptic integral of the first kind of real or complex numbers

carlsonRG( x, y, z ) — Carlson completely symmetric elliptic integral of the second kind of real or complex numbers

carlsonRJ( x, y, z, w ) — Carlson symmetric elliptic integral of the third kind of real or complex numbers


Elliptic Functions 椭圆函数

jacobiTheta( n, x, q ) — Jacobi theta function n of a real or complex number with real or complex nome q

ellipticNome( m ) — elliptic nome q of a real or complex elliptic parameter m

am( x, m ) — Jacobi amplitude of a real or complex number with real or complex elliptic parameter m

sn( x, m ) — Jacobi elliptic sine of a real or complex number with real or complex elliptic parameter m

cn( x, m ) — Jacobi elliptic cosine of a real or complex number with real or complex elliptic parameter m

dn( x, m ) — Jacobi delta amplitude of a real or complex number with real or complex elliptic parameter m

weierstrassRoots( g2, g3 ) — Weierstrass roots e1, e2 and e3 for real or complex invariants. Returned as an array.

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.

weierstrassInvariants( w1, w3 ) — Weierstrass invariants g2 and g3 for real or complex half periods. Returned as an array.

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.

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.

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.

kleinJ( x ) — Klein j-invariant of a complex number


Hypergeometric Functions 超几何函数

hypergeometric0F1( a, x ) — confluent hypergeometric function of a real or complex parameter a of a real or complex number

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

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

whittakerM( k, m, x ) — Whittaker function of the first kind of real or complex parameters k and m of a real or complex number

whittakerW( k, m, x ) — Whittaker function of the second kind of real or complex parameters k and m of a real or complex number

hypergeometric2F1( a, b, c, x ) — Gauss hypergeometric function of real or complex parameters a, b and c of a real or complex number

hypergeometric1F2( a, b, c, x ) — hypergeometric function of real or complex parameters a, b and c of a real or complex number

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

factorial( n ) — factorial of a real or complex number

factorial2( n ) — double factorial of a real or complex number

binomial( n, m ) — binomial coefficient of real or complex numbers

logGamma( x ) — logarithm of the gamma function of a real or complex number

gamma( x ) — gamma function of a real or complex number

gamma( x, y ) — upper incomplete gamma function of real or complex numbers

gamma( x, 0, y ) — lower incomplete gamma function of real or complex numbers

gamma( x, y, z ) — generalized incomplete gamma function of real or complex numbers

beta( x, y ) — beta function of real or complex numbers

beta( x, y, z ) — incomplete beta function of real or complex numbers, where x = 1 replicates the beta function

psi(x)=digamma( x ) — digamma function of a real or complex number


Gamma-Type Functions

erf( x ) — error function of a real or complex number

erfc( x ) — complementary error function of a real or complex number

erfi( x ) — imaginary error function of a real or complex number

fresnelS( x ) — Fresnel sine integral of a real or complex number

fresnelC( x ) — Fresnel cosine integral of a real or complex number

Ei(x)=expIntegral( x ) — exponential integral of a real or complex number

li(x)=logIntegral( x ) — logarithmic integral of a real or complex number

si(x)=sinIntegral( x ) — sine integral of a real or complex number

ci(x)cosIntegral( x ) — cosine integral of a real or complex number

shi(x)=>sinhIntegral( x ) — hyperbolic sine integral of a real or complex number

chi(x=)coshIntegral( x ) — hyperbolic cosine integral of a real or complex number

En(n,x)=expIntegralE( n, x ) — generalized exponential integral of a real or complex order n of a real or complex number


Zeta Functions

zeta( x ) — Riemann zeta of a real or complex number

eta(x)=dirichletEta( x ) — Dirichlet eta of a real or complex number

bernoulli( n ) — Bernoulli number for index n

harmonic( n ) — harmonic number for index n

hurwitzZeta( x, a ) — Hurwitz zeta function of a real or complex number with real or complex parameter a


Miscellaneous Functions

chop( x ) — set real and complex parts smaller than 10−10 to zero

chop( x, tolerance ) — set real and complex parts smaller than tolerance to zero

kronecker( i, j ) — Kronecker delta δij for integer arguments

piecewise( [ function, [begin,end] ], … ) — piecewise expression defined on an arbitrary number of subdomains returned as a function


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