# LogBarnesG

LogBarnesG[z]

gives the logarithm of the Barnes G-function .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• LogBarnesG[z] is analytic throughout the complex z plane.
• LogBarnesG[z] is analytic throughout the complex z plane and is defined as .
• For certain special arguments, LogBarnesG automatically evaluates to exact values.
• LogBarnesG can be evaluated to arbitrary numerical precision.
• LogBarnesG automatically threads over lists.

# Examples

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## Basic Examples(6)

Evaluate numerically:

Evaluate at large arguments:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

Series expansion at a singular point:

## Scope(21)

### Numerical Evaluation(4)

Evaluate numerically:

Evaluate to high precision:

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

Complex number inputs:

Evaluate efficiently at high precision:

### Specific Values(5)

Values at fixed points:

Value at infinity:

Value at zero:

Evaluate symbolically:

Find the first positive maximum:

### Visualization(2)

Plot the LogBarnesG function:

Plot the real part of :

Plot the imaginary part of :

### Function Properties(3)

Real domain of LogBarnesG:

Complex domain:

Function range of LogBarnesG:

### Differentiation(3)

First derivatives with respect to z:

Higher derivatives with respect to z:

Plot the higher derivatives with respect to z:

Formula for the  derivative with respect to z:

### Series Expansions(4)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Find the series expansion at Infinity:

Find series expansion for an arbitrary symbolic direction :

Taylor expansion at a generic point:

## Generalizations & Extensions(1)

LogBarnesG can be applied to a power series:

## Applications(1)

Concavity property of BarnesG:

## Properties & Relations(2)

Use FunctionExpand to expand LogBarnesG in terms of related polygamma functions:

LogBarnesG is the sum of LogGamma functions:

Introduced in 2008
(7.0)