HypergeometricPFQRegularized

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

is the regularized generalized hypergeometric function .

Details

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  (24)

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 _1F^~_2(1/2;1/2,1/3;z):

Plot the imaginary part of _1F^~_2(1/2;1/2,1/3;z):

Function Properties  (3)

HypergeometricPFQRegularized is defined for all real and complex values:

HypergeometricPFQRegularized threads elementwise over lists in its third argument:

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 ^(th) 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:

Introduced in 1996
 (3.0)