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Special Functions Available in MuPAD

Dirac and Heaviside Functions

The following MuPAD® functions represent the Dirac δ-distribution and the Heaviside (step) function. You can use these functions as input for your computations. MuPAD can return results in terms of these functions:

dirac

Dirac δ-function

heaviside

Heaviside function

Gamma Functions

MuPAD provides the following functions to represent the β-function, Γ-function, and other related special functions. You can use these functions as input for your computations. MuPAD can return results in terms of these functions:

beta

β-function

binomial

Binomial expression

gamma

Γ-function

igamma

Incomplete Γ-function

lngamma

Logarithmic Γ-function

psi

Polygamma function

Zeta Function and Polylogarithms

The following MuPAD functions represent the Riemann ζ-function and the related dilogarithm and polylogarithm special functions. You can use these functions as input for your computations. MuPAD can return results in terms of these functions:

dilog

Dilogarithm function

polylog

Polylogarithm function

zeta

Riemann ζ-function

Airy and Bessel Functions

The following MuPAD functions represent the Bessel and Airy special functions. You can use these functions as an input for your computations. MuPAD can return results in terms of these functions:

airyAi

Airy function Ai

airyBi

Airy function Bi

besselI

Modified Bessel function of the first kind

besselJ

Bessel function of the first kind

besselK

Modified Bessel function of the second kind

besselY

Bessel function of the second kind

Exponential and Trigonometric Integrals

The following MuPAD functions represent exponential and trigonometric integrals. You can use these functions as an input for your computations. MuPAD can return results in terms of these functions:

Ci

Cosine integral

Chi

Hyperbolic cosine integral

Ei

Exponential integral

Si

Sine integral

Ssi

Shifted sine integral

Shi

Hyperbolic sine integral

Error Functions and Fresnel Functions

The following MuPAD functions represent the error functions (integrals of Gaussian distribution) and Fresnel functions. You can use these functions as input for your computations. MuPAD can return results in terms of these functions:

erf

Error function

erfc

Complementary error function

erfi

Imaginary error function

inverf

Inverse of error function

inverfc

Inverse of complementary error function

fresnelC

Fresnel cosine integral function

fresnelS

Fresnel sine integral function

Hypergeometric, Meijer G, and Whittaker Functions

The following MuPAD functions represent the hypergeometric function, the more general Meijer G function, and related functions. You can use these functions as input for your computations. MuPAD can return results in terms of these functions:

hypergeom

Hypergeometric function

kummerU

Confluent hypergeometric KummerU function

meijerG

Meijer G function

whittakerM

Whittaker's M function

whittakerW

Whittaker's W function

Elliptic Integrals

The following MuPAD functions represent the elliptic integrals of different kinds. You can use these functions as input for your computations. MuPAD can return results in terms of these functions:

ellipticK

Complete elliptic integral of the first kind

ellipticCK

Complementary complete elliptic integral of the first kind

ellipticF

Incomplete elliptic integral of the first kind

ellipticE

Elliptic integral of the second kind

ellipticCE

Complementary complete elliptic integral of the second kind

ellipticPi

Elliptic integral of the third kind

ellipticCPi

Complementary complete elliptic integral of the third kind

ellipticNome

Elliptic nome

Lambert W Function (omega Function)

The lambertW function represents the solutions of the equation yey = x. You can use the function as input for your computations. MuPAD can return results in terms of this function.

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