# 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(33)

### Numerical Evaluation(5)

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:

MeijerG threads elementwise over lists in its third argument:

MeijerG threads elementwise over sparse and structured arrays in its third argument:

### 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},{}},z ]:

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

### 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(5)

Define the product of independent random variables drawn from BetaDistribution:

The PDF of the distribution is defined in terms of MeijerG:

Use FunctionExpand to express it in terms of simpler functions:

Compare the plot of the PDF to the Histogram of a 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:

A formula for solutions to the trinomial equation :

First root of the quintic :

Check the solution:

## 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:

Wolfram Research (1996), MeijerG, Wolfram Language function, https://reference.wolfram.com/language/ref/MeijerG.html.

#### Text

Wolfram Research (1996), MeijerG, Wolfram Language function, https://reference.wolfram.com/language/ref/MeijerG.html.

#### CMS

Wolfram Language. 1996. "MeijerG." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/MeijerG.html.

#### APA

Wolfram Language. (1996). MeijerG. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MeijerG.html

#### BibTeX

@misc{reference.wolfram_2024_meijerg, author="Wolfram Research", title="{MeijerG}", year="1996", howpublished="\url{https://reference.wolfram.com/language/ref/MeijerG.html}", note=[Accessed: 20-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_meijerg, organization={Wolfram Research}, title={MeijerG}, year={1996}, url={https://reference.wolfram.com/language/ref/MeijerG.html}, note=[Accessed: 20-July-2024 ]}