Built-in Mathematica Symbol

HypergeometricPFQ

HypergeometricPFQ[{a₁, ..., a_p}, {b₁, ..., b_q}, z]
is the generalized hypergeometric function

.

MORE INFORMATION

Mathematical function, suitable for both symbolic and numerical manipulation.

has series expansion .

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 (4)

Evaluate numerically:

Plot

:

Evaluate symbolically:

Series at the origin:

Evaluate numerically:

In[1]:=

Out[1]=

Plot

:

In[1]:=

Out[1]=

Evaluate symbolically:

In[1]:=

Out[1]=

Series at the origin:

In[1]:=

Out[1]=

Scope (7)

Evaluate for complex arguments and parameters:

Evaluate to high precision:

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

HypergeometricPFQ threads element-wise over lists in its third argument:

For simple parameters, HypergeometricPFQ evaluates to simpler functions:

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

TraditionalForm formatting:

Generalizations & Extensions (2)

Expand HypergeometricPFQ of type

into a series at the branch point

:

Expand HypergeometricPFQ into a series around

:

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: