HypergeometricPFQ

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

is the generalized hypergeometric function .

Details

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  (32)

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  (9)

Domain of HypergeometricPFQ:

Permutation symmetry:

HypergeometricPFQ is an analytic function of z for specific values:

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

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

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

HypergeometricPFQ is neither non-negative nor non-positive:

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

HypergeometricPFQ is neither convex nor concave:

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:

TraditionalForm formatting:

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:

Wolfram Research (1996), HypergeometricPFQ, Wolfram Language function, https://reference.wolfram.com/language/ref/HypergeometricPFQ.html (updated 1999).

Text

Wolfram Research (1996), HypergeometricPFQ, Wolfram Language function, https://reference.wolfram.com/language/ref/HypergeometricPFQ.html (updated 1999).

BibTeX

@misc{reference.wolfram_2021_hypergeometricpfq, author="Wolfram Research", title="{HypergeometricPFQ}", year="1999", howpublished="\url{https://reference.wolfram.com/language/ref/HypergeometricPFQ.html}", note=[Accessed: 18-September-2021 ]}

BibLaTeX

@online{reference.wolfram_2021_hypergeometricpfq, organization={Wolfram Research}, title={HypergeometricPFQ}, year={1999}, url={https://reference.wolfram.com/language/ref/HypergeometricPFQ.html}, note=[Accessed: 18-September-2021 ]}

CMS

Wolfram Language. 1996. "HypergeometricPFQ." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 1999. https://reference.wolfram.com/language/ref/HypergeometricPFQ.html.

APA

Wolfram Language. (1996). HypergeometricPFQ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HypergeometricPFQ.html