This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# PolyGamma

 PolyGamma[z]gives the digamma function . PolyGammagives the n derivative of the digamma function .
• Mathematical function, suitable for both symbolic and numerical manipulation.
• PolyGamma[z] is the logarithmic derivative of the gamma function, given by .
• PolyGamma is given for positive integer by .
• For arbitrary complex n the polygamma function is defined by fractional calculus analytic continuation.
• PolyGamma[z] and PolyGamma 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.
Evaluate the digamma function:
Derivative of the gamma function:
The digamma function:
Evaluate the digamma function:
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Derivative of the gamma function:
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The digamma function:
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 Scope   (9)
Evaluate for integer arguments of any size:
Evaluate numerically:
Evaluate for complex arguments and orders:
Evaluate to any precision:
The precision of the output tracks the precision of the input:
Use FunctionExpand to expand higher-order polygamma functions:
Series expansion:
Infinite arguments can give symbolic results:
PolyGamma can be applied to a power series:
Series expansion at poles:
Series expansion at infinities:
Special cases:
Evaluate at exact arguments:
Series expansion at a generic point:
Series expansion near a singularity:
 Applications   (3)
Plot of the absolute value of PolyGamma over the complex plane:
The electric field energy of a charge at a fraction of the distance between parallel conducting plates:
Expand near the left wall:
Final speed of a rocket with discrete propulsion events:
Final velocity in the limit of constant continuous propulsion:
Use FullSimplify to simplify polygamma functions:
Express rational arguments through elementary functions:
Numerically find a root of a transcendental equation:
Sums and integrals:
Generate PolyGamma from integrals, sums, and limits:
Generating function:
Obtain as special cases of hypergeometric functions:
The one-argument form evaluates to the two-argument form:
Large orders can give results too large to be computed explicitly:
Machine-number inputs can give high-precision results: