# HypergeometricPFQ

HypergeometricPFQ[{a1,,ap},{b1,,bq},z]

is the generalized hypergeometric function .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• has series expansion , where is the Pochhammer symbol.
• Hypergeometric0F1, Hypergeometric1F1, and Hypergeometric2F1 are special cases of HypergeometricPFQ.
• In many special cases, HypergeometricPFQ is automatically converted to other functions.
• For certain special arguments, HypergeometricPFQ automatically evaluates to exact values.
• HypergeometricPFQ can be evaluated to arbitrary numerical precision.
• For , HypergeometricPFQ[alist,blist,z] has a branch cut discontinuity in the complex plane running from to .
• FullSimplify and FunctionExpand include transformation rules for HypergeometricPFQ.

# Examples

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## Basic Examples(5)

Evaluate numerically:

Plot over a subset of the reals:

Evaluate symbolically:

Series expansion at the origin:

Series expansion at Infinity:

## Scope(25)

### Numerical Evaluation(4)

Evaluate to high precision:

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

Evaluate for complex arguments and parameters:

Evaluate HypergeometricPFQ efficiently at high precision:

HypergeometricPFQ threads elementwise over lists in its third argument:

### Specific Values(4)

For simple parameters, HypergeometricPFQ evaluates to simpler functions:

HypergeometricPFQ evaluates to a polynomial if any of the parameters ak is a non-positive integer:

Value at the origin:

Find a value of satisfying the equation :

### Visualization(2)

Plot the HypergeometricPFQ function:

Plot the real part of :

Plot the imaginary part of :

### Function Properties(2)

Domain of HypergeometricPFQ:

Permutation symmetry:

### Differentiation(2)

First derivative:

Higher derivatives:

Plot higher derivatives for some values parameters:

### Integration(3)

Indefinite integral of HypergeometricPFQ:

Definite integral of HypergeometricPFQ:

Integral with a power function:

### Series Expansions(4)

Taylor expansion for HypergeometricPFQ:

Plot the first three approximations for around :

General term in the series expansion of HypergeometricPFQ:

Expand HypergeometricPFQ of type into a series at the branch point :

Expand HypergeometricPFQ into a series around :

### Function Representations(4)

Primary definition:

HypergeometricPFQ can be represented as a DifferentialRoot:

HypergeometricPFQ can be represented in terms of MeijerG:

## Applications(5)

Solve a differential equation of hypergeometric type:

A formula for solutions to trinomial equation :

First root of the quintic :

Check the solution:

Effective confining potential in random matrix theory for a Gaussian density of states:

Expansion at infinity reveals logarithmic growth:

Surface tension of an electrolyte solution as a function of concentration y:

Onsager law for small concentrations:

Fractional derivative of Sin:

Derivative of order of Sin:

Plot a smooth transition between the derivative and integral of Sin:

## Properties & Relations(3)

Integrate frequently returns results containing HypergeometricPFQ:

Sum may return results containing HypergeometricPFQ:

Use FunctionExpand to transform HypergeometricPFQ into less general functions:

## Possible Issues(2)

Machine-precision input may be insufficient to get a correct answer:

With exact input, the answer is correct:

Common symbolic parameters in HypergeometricPFQ generically cancel:

However, when there is a negative integer among common elements, HypergeometricPFQ is interpreted as a polynomial:

## Neat Examples(1)

The period of an anharmonic oscillator with Hamiltonian :

Period for quartic anharmonicity:

Limit of pure quartic potential:

Introduced in 1996
(3.0)
|
Updated in 1999
(4.0)