# FoxH FoxH[{{{a1,α1},,{an,αn}},{{an+1,αn+1},,{ap,αp}}},{{{b1,β1},,{bm,βm}},{{bm+1,βm+1},,{bq,βq}}},z]

is the Fox H-function .

# Details  • Mathematical function, suitable for both symbolic and numerical manipulation.
• FoxH generalizes the MeijerG function and is defined by the MellinBarnes integral where and are positive real numbers and the integration is along a path separating the poles of from the poles of .
• Three choices are possible for the path :
a. is a loop beginning at and ending at and encircling all the poles of once in the positive direction.
b. is a loop beginning at and ending at and encircling all the poles of once in the negative direction.
c. is a contour starting at the point and going to such that all the poles of are separated from the poles of .
• • FoxH specializes to MeijerG if for and : .
• In many special cases, FoxH is automatically converted to other functions.
• FoxH can be evaluated for arbitrary complex parameters.
• FoxH can be evaluated to arbitrary numerical precision.
• FoxH automatically threads over lists.

# Examples

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## Basic Examples(6)

Evaluate numerically:

Many special functions are special cases of FoxH:

Some cases cannot be easily expressed in terms of MeijerG:

Plot the FoxH function:

Plot over a subset of the complexes:

Series expansion at the origin:

## Scope(26)

### Numerical Evaluation(5)

Evaluate to high precision:

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

FoxH takes complex number parameters and :

FoxH takes complex number arguments:

Evaluate FoxH efficiently at high precision:

### Specific Values(3)

Values at fixed points:

Evaluate FoxH symbolically:

Values at zero:

### Visualization(4)

Plot a family of FoxH functions:

ComplexContourPlot of FoxH[{{{},{}},{{{-1,1/2}},{}}, z]:

Use AbsArgPlot and ReImPlot to plot complex values of FoxH over the real numbers:

Plot FoxH as a function of parameters and :

### Function Properties(5)

For simple parameters, FoxH evaluates to simpler functions:

FoxH is symmetric in the pairs and :

FoxH might reduce to a simpler FoxH if some of the pairs are equal:

A more complex case:

FoxH threads elementwise over lists in the last argument:

### Differentiation(2)

First derivative with respect to z:

Higher-order derivative with respect to z:

Formula for the  derivative of a specific FoxH with respect to z:

### Integration(3)

Compute the indefinite integral using Integrate:

Verify it by calculating the antiderivative:

Definite integral:

Some other integrals:

### Series Expansions(4)

Get the series expansion of some FoxH function at the origin:

The first three approximations of this FoxH function around :

Plot these approximations:

Find the series expansion of a general FoxH function at the origin:

Find the series expansion of a general FoxH function at Infinity:

Get the general term in the series expansion using SeriesCoefficient:

## Applications(3)

Use FoxHReduce to get the representation of almost any mathematical function in terms of FoxH:

More exotic combinations:

A root of the trinomial equation can be written in terms of FoxH:

Verify this for :

The roots of the general trinomial can also be expressed in terms of FoxH:

Verify these roots for and :

Express the PDF of StableDistribution in terms of FoxH for the case of :

Evaluate it and compare with the built-in PDF generated using StableDistribution:

## Properties & Relations(2)

Use FunctionExpand to expand FoxH into simpler functions:

FoxHReduce returns FoxH representations of functions:

## Possible Issues(3) is a singular point of FoxH:

Series at the origin is available when the parameters are positive and are positive and exact:

Series at Infinity is available when the parameters are positive and are positive and exact:

## Neat Examples(1)

Many elementary and special functions are special cases of FoxH: