Documentation

## Special Functions Available in MuPAD

MuPAD® notebooks will be removed in a future release. Use MATLAB® live scripts instead.

MATLAB live scripts support most MuPAD functionality, though there are some differences. For more information, see Convert MuPAD Notebooks to MATLAB Live Scripts.

### 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.

#### Mathematical Modeling with Symbolic Math Toolbox

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