Evaluate numerically to high precision:

The precision of the output tracks the precision of the input:

Complex arguments:

Evaluate LogGamma efficiently at high precision:

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Or compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix LogGamma function using MatrixFunction:

Give exact results for integers and half-integers:

Some singular points of LogGamma:

Values at infinity:

Find a zero of LogGamma:

Plot LogGamma:

Plot the real part of :

Plot the imaginary part of :

LogGamma is defined for all positive real values:

Complex domain:

LogGamma is not an analytic function:

Has both singularities and discontinuities:

LogGamma is neither nondecreasing nor nonincreasing:

LogGamma is not injective:

LogGamma is not surjective:

LogGamma is neither non-negative nor non-positive:

LogGamma is convex on its real domain:

First derivative of LogGamma:

Higher derivatives of LogGamma:

Formula for the derivative:

Indefinite integral of LogGamma:

Compute integrals involving LogGamma:

Definite integral for complex values of LogGamma:

Series expansion at the origin:

Taylor expansion for LogGamma around :

Plot the first three approximations for LogGamma around :

Series expansion at infinity:

Give the result for an arbitrary symbolic direction :

Series expansion at poles of the LogGamma function:

LogGamma can be applied to a power series:

Use FullSimplify to simplify logarithmic gamma functions:

Use FunctionExpand to express through Gamma:

Recurrence relation:

Series representation of LogGamma:

Integral representation of LogGamma through PolyGamma:

TraditionalForm formatting: