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

# LogGamma

 LogGamma[z]gives the logarithm of the gamma function .
• Mathematical function, suitable for both symbolic and numerical manipulation.
• LogGamma[z] is analytic throughout the complex z plane, except for a single branch cut discontinuity along the negative real axis. Log[Gamma[z]] has a more complex branch cut structure.
• For certain special arguments, LogGamma automatically evaluates to exact values.
• LogGamma can be evaluated to arbitrary numerical precision.
• LogGamma automatically threads over lists.
Evaluate numerically:
Evaluate at large arguments:
Evaluate numerically:
 Out[1]=
Evaluate at large arguments:
 Out[2]=

 Out[1]=
 Scope   (7)
Give exact results for integers and half-integers:
Complex arguments:
Evaluate to high precision:
The precision of the output tracks the precision of the input:
Series expansion at the origin:
Series expansion at infinity:
Give the result for an arbitrary symbolic direction :
Infinite arguments give symbolic results:
LogGamma can be applied to a power series:
Series expansion at poles of the LogGamma function:
 Applications   (3)
Plot of the imaginary part of LogGamma[z] and Log over the complex plane:
Calculate ratio of Gamma functions at very large arguments:
Direct calculation fails because intermediate numbers are too large:
Find the first few digits of :
Use FullSimplify to simplify logarithmic gamma functions:
Use FunctionExpand to express through Gamma:
Numerically find a root of a transcendental equation:
Integrals:
In TraditionalForm, is automatically interpreted as the gamma function:
For many complex values :
Algorithmically generated results typically contain instead of :
Plot LogGamma at the Gaussian integers:
Riemann surface of LogGamma:
New in 2