# LogGamma LogGamma[z]

gives the logarithm of the gamma function .

# Details • 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.
• LogGamma can be used with Interval and CenteredInterval objects. »

# Examples

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## Basic Examples(5)

Evaluate numerically:

Evaluate at large arguments:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at Infinity:

## Scope(36)

### Numerical Evaluation(5)

Evaluate numerically to high precision:

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

Complex arguments:

Evaluate LogGamma efficiently at high precision:

LogGamma can be used with Interval and CenteredInterval objects:

### Specific Values(4)

Give exact results for integers and half-integers:

Some singular points of LogGamma:

Values at infinity:

Find a zero of LogGamma:

### Visualization(2)

Plot LogGamma:

Plot the real part of :

Plot the imaginary part of :

### Function Properties(8)

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:

### Differentiation(3)

First derivative of LogGamma:

Higher derivatives of LogGamma:

Formula for the  derivative:

### Integration(3)

Indefinite integral of LogGamma:

Compute integrals involving LogGamma:

Definite integral for complex values of LogGamma:

### Series Expansions(5)

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:

### Function Identities and Simplifications(3)

Use FullSimplify to simplify logarithmic gamma functions:

Use FunctionExpand to express through Gamma:

Recurrence relation:

### Function Representations(3)

Series representation of LogGamma:

Integral representation of LogGamma through PolyGamma:

## Applications(4)

Calculate ratio of Gamma functions at very large arguments:

Direct calculation fails because intermediate numbers are too large:  Find the first few digits of :

Plot of the imaginary part of LogGamma[z] and Log[Gamma[z]] over the complex plane:

Determine the number of bins to use for bimodal data by Knuth's Bayesian method:

The optimal number of bins maximizes the log of the posterior density:

## Properties & Relations(6)

Use FullSimplify to simplify logarithmic gamma functions:

Use FunctionExpand to express through Gamma:

Numerically find a root of a transcendental equation:

Check that is convex for :

Find minimum of on the positive real axis:

Visualize the result:

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: