Built-in Mathematica Symbol

LogGamma

LogGamma[z]
gives the logarithm of the gamma function

.

MORE INFORMATION

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.

EXAMPLES

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Basic Examples (2)

Evaluate numerically:

Evaluate at large arguments:

Evaluate numerically:

In[1]:=

Out[1]=

Evaluate at large arguments:

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

:

TraditionalForm formatting:

Generalizations & Extensions (4)

Infinite arguments give symbolic results:

LogGamma threads element-wise over lists:

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[Gamma[z]] 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

:

Properties & Relations (5)

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:

Possible Issues (2)

For many complex values

:

Algorithmically generated results typically contain

instead of

:

Neat Examples (2)

Plot LogGamma at the Gaussian integers:

Riemann surface of LogGamma: