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LogGamma

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

LogGamma[z]
    gives the log-gamma function log(Gamma(z)), analytic except for a branch
cut on the negative reals. Exact at integer and half-integer z (with the
negative-axis branch term), divergent (Infinity) at non-positive integers,
and evaluated numerically for real and complex z at machine or arbitrary
(MPFR) precision. D[LogGamma[z], z] is PolyGamma[0, z]. Listable.

Examples

No verified examples yet for this function.

Implementation notes

  • Exact closed forms. Integers reduce as LogGamma[n] = Log[(n-1)!]

Attributes: Listable, NumericFunction, Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= LogGamma[5]
Out[1]= Log[24]

In[1]:= LogGamma[1/2]
Out[1]= Log[Sqrt[Pi]]

In[1]:= D[LogGamma[z], z]
Out[1]= PolyGamma[0, z]

In[1]:= N[LogGamma[100], 40]
Out[1]= 359.13420536957539877604401046028690961264

In[1]:= N[LogGamma[1 + I], 30]
Out[1]= -0.6509231993018563388852168315042 - 0.3016403204675331978875316577968*I

Notes

LogGamma[z] is log(Gamma(z)), analytic except for a branch cut on the negative reals. It is exact at integer and half-integer arguments (LogGamma[5] is Log[4!] = Log[24]), divergent at non-positive integers, and evaluates numerically for real or complex z at machine or arbitrary (MPFR) precision. Its derivative is PolyGamma[0, z]. Unlike Log[Gamma[z]], LogGamma tracks the correct sheet, which matters for large or complex arguments where Gamma overflows. Listable.