# HypergeometricPFQRegularized

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

is the regularized generalized hypergeometric function .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• HypergeometricPFQRegularized is finite for all finite values of its arguments so long as .
• For certain special arguments, HypergeometricPFQRegularized automatically evaluates to exact values.
• HypergeometricPFQRegularized can be evaluated to arbitrary numerical precision.

# Examples

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## Basic Examples(6)

Evaluate numerically:

Evaluate symbolically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

## Scope(31)

### Numerical Evaluation(4)

Evaluate numerically:

Evaluate to high precision:

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

Complex number inputs:

Evaluate efficiently at high precision:

### Specific Values(4)

For simple parameters, HypergeometricPFQRegularized evaluates to simpler functions:

Evaluate symbolically:

Value at zero:

Find a value of for which HypergeometricPFQRegularized[{2,1},{2,3},x]1.5:

### Visualization(2)

Plot the HypergeometricPFQRegularized function for various parameters:

Plot the real part of :

Plot the imaginary part of :

### Function Properties(10)

HypergeometricPFQRegularized is defined for all real and complex values:

HypergeometricPFQRegularized threads elementwise over lists in its third argument:

HypergeometricPFQRegularized is an analytic function of z for specific values:

HypergeometricPFQRegularized is neither non-decreasing nor non-increasing for specific values:

HypergeometricPFQRegularized[{1,1,1},{3,3,3},z] is injective:

HypergeometricPFQRegularized[{1,1,1},{3,3,3},z] is not surjective:

HypergeometricPFQRegularized is neither non-negative nor non-positive:

HypergeometricPFQRegularized[{1,1,2},{3,3},z] has both singularity and discontinuity for z1 and at zero:

HypergeometricPFQRegularized is neither convex nor concave:

TraditionalForm formatting:

### Differentiation(3)

First derivative with respect to :

Higher derivatives with respect to :

Plot the higher derivatives with respect to when and :

Formula for the  derivative with respect to z when a1=1,a2=2 and b1=b2=b3=3:

### Integration(3)

Compute the indefinite integral using Integrate:

Verify the antiderivative:

Definite integral:

More integrals:

### Series Expansions(5)

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:

Find the series expansion for an arbitrary symbolic direction :

Taylor expansion at a generic point:

## Applications(1)

Find a fractional derivative of BesselJ:

Integral of order of BesselJ[0,z]:

## Properties & Relations(2)

Use FunctionExpand to express the input in terms of simpler functions:

Integrate may return results involving HypergeometricPFQRegularized: