# MeijerG

MeijerG[{{a1,,an},{an+1,,ap}},{{b1,,bm},{bm+1,,bq}},z]

is the Meijer G-function .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• The generalized form MeijerG[alist,blist,z,r] is defined for real r by , where in the default case .
• In many special cases, MeijerG is automatically converted to other functions.

# Examples

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## Basic Examples(6)

Evaluate numerically:

Many special functions are special cases of MeijerG:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

## Scope(32)

### Numerical Evaluation(4)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number input:

Evaluate efficiently at high precision:

### Specific Values(5)

Values at fixed points:

Evaluate symbolically:

Values at zero:

For simple parameters, MeijerG evaluates to simpler functions:

Find a positive minimum of MeijerG[{{},{}},{{1/2},{3/2}},x]:

### Visualization(2)

Plot the MeijerG function for various parameters:

Plot the real part of MeijerG[{{1},{}},{{1/2,1,3/2},{}},x+ y ]:

Plot the imaginary part of MeijerG[{{1},{}},{{1/2,1,3/2},{}},x+ y ]:

### Function Properties(9)

Real and complex domains of :

MeijerG threads elementwise over lists in the last argument: is not an analytic function:

Has both singularities and discontinuities: is nonincreasing over its real domain: is injective: is not surjective: is negative over its real domain: is convex over its real domain:

### Differentiation(3)

First derivative with respect to z:

Higher derivatives with respect to z:

Plot the higher derivatives with respect to z when b=3 and c=2:

Formula for the  derivative with respect to z:

### Integration(3)

Compute the indefinite integral using Integrate:

Verify the antiderivative:

Definite integral:

More integrals:

### Series Expansions(6)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

General term in the series expansion using SeriesCoefficient:

Find the series expansion at Infinity:

Series expansion in a logarithmic case:

Taylor expansion at a generic point:

## Generalizations & Extensions(1)

Evaluate a generalized Meijer G function:

The analogous ordinary Meijer G function has a different branch cut structure:

## Applications(4)

Give PDF of a product of independent random variables from BetaDistribution:

Using FunctionExpand to express it in terms of simpler functions:

Compare the plot of the PDF to the Histogram of random sample:

Solve a differential equation:

MeijerG gives a logarithmic part:

Integrate can return answers involving MeijerG:

Solve a third-order singular ODE in terms of the HypergeometricPFQ and MeijerG functions:

Verify that the components of the general solution for an ODE are linearly independent:

## Properties & Relations(1)

Use FunctionExpand to expand MeijerG into simpler functions:

## Possible Issues(3)

For some choices of parameters, MeijerG is not defined:  is a singular point of MeijerG functions with :

MeijerG is a piecewise analytic function for :

## Neat Examples(2)

Solve a SIAM 100digit challenge problem: find to maximize:

Plot the integral:

Numerically find the maximum:

Generate many elementary and special functions as special cases of MeijerG: