# Wolfram Language & System 10.4 (2016)|Legacy Documentation

This is documentation for an earlier version of the Wolfram Language.
BUILT-IN WOLFRAM LANGUAGE SYMBOL

# PolyGamma

PolyGamma[z]
gives the digamma function .

PolyGamma[n,z]
gives the n derivative of the digamma function .

## DetailsDetails

• Mathematical function, suitable for both symbolic and numerical manipulation.
• PolyGamma[z] is the logarithmic derivative of the gamma function, given by .
• PolyGamma[n,z] is given for positive integer by .
• For arbitrary complex n the polygamma function is defined by fractional calculus analytic continuation.
• PolyGamma[z] and PolyGamma[n,z] are meromorphic functions of z with no branch cut discontinuities.
• For certain special arguments, PolyGamma automatically evaluates to exact values.
• PolyGamma can be evaluated to arbitrary numerical precision.
• PolyGamma automatically threads over lists.

## ExamplesExamplesopen allclose all

### Basic Examples  (3)Basic Examples  (3)

Evaluate the digamma function:

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Derivative of the gamma function:

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The digamma function:

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