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# HypergeometricPFQ

 HypergeometricPFQis the generalized hypergeometric function .
• Mathematical function, suitable for both symbolic and numerical manipulation.
• has series expansion .
• In many special cases, HypergeometricPFQ is automatically converted to other functions.
• For certain special arguments, HypergeometricPFQ automatically evaluates to exact values.
• For , HypergeometricPFQ has a branch cut discontinuity in the complex plane running from to .
Evaluate numerically:
Plot :
Evaluate symbolically:
Series at the origin:
Evaluate numerically:
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Plot :
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Evaluate symbolically:
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Series at the origin:
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 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 is a non-positive integer:
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 :
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:
Integrate frequently returns results containing HypergeometricPFQ:
Sum may return results containing HypergeometricPFQ:
Use FunctionExpand to transform HypergeometricPFQ into less general functions:
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:
The period of an anharmonic oscillator with Hamiltonian :
Period for quartic anharmonicity:
Limit of pure quartic potential:
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