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  (26)

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  (3)

Real and complex domains of MeijerG:

MeijerG threads elementwise over lists in the last argument:

TraditionalForm formatting:

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 ^(th) 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  (3)

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:

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:

Introduced in 1996
 (3.0)