Function Repository Resource:

TripleGamma

Evaluate the triple gamma function

Contributed by: Jan Mangaldan
 ResourceFunction["TripleGamma"][z] gives the triple gamma function .

Details

Mathematical function, suitable for both symbolic and numeric manipulation.
The triple gamma function is defined as for positive integers n and through analytic continuation elsewhere.
The triple gamma function satisfies the functional equation , where is is the BarnesG function.
ResourceFunction["TripleGamma"][z] has no branch cut discontinuities.
For integers and half-integers, ResourceFunction["TripleGamma"] automatically evaluates to exact values.
ResourceFunction["TripleGamma"] can be evaluated to arbitrary numerical precision.

Examples

Basic Examples (4)

Evaluate triple gamma on integer values:

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Half-integer values:

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Evaluate numerically for complex arguments:

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Plot over a subset of the reals:

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Scope (4)

Evaluate numerically:

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Evaluate to arbitrary precision:

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

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Evaluate for complex argument:

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Applications (1)

Plot of the absolute value of TripleGamma in the complex plane:

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Properties and Relations (2)

Compare TripleGamma with an explicit product representation for integer argument:

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TripleGamma satisfies a functional equation involving BarnesG:

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Version History

• 1.0.0 – 09 March 2021