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

# 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(27)

### Numerical Evaluation(5)

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:

LogBarnesG can be used with Interval and CenteredInterval objects:

### Specific Values(4)

Values at fixed points:

Value at infinity:

Value at zero:

Find the first positive maximum:

### Visualization(2)

Plot the LogBarnesG function:

Plot the real part of :

Plot the imaginary part of :

### Function Properties(9)

Real domain of LogBarnesG:

Complex domain:

Function range of LogBarnesG:

LogBarnesG is not an analytic function of x:

LogBarnesG is neither non-increasing nor non-decreasing:

LogBarnesG is not injective:

LogBarnesG is surjective:

LogBarnesG is neither non-negative nor non-positive:

LogBarnesG has both singularities and discontinuities for negative values:

LogBarnesG is neither convex nor concave:

### 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(1)

LogBarnesG is the sum of LogGamma functions:

Wolfram Research (2008), LogBarnesG, Wolfram Language function, https://reference.wolfram.com/language/ref/LogBarnesG.html (updated 2022).

#### Text

Wolfram Research (2008), LogBarnesG, Wolfram Language function, https://reference.wolfram.com/language/ref/LogBarnesG.html (updated 2022).

#### CMS

Wolfram Language. 2008. "LogBarnesG." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/LogBarnesG.html.

#### APA

Wolfram Language. (2008). LogBarnesG. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LogBarnesG.html

#### BibTeX

@misc{reference.wolfram_2024_logbarnesg, author="Wolfram Research", title="{LogBarnesG}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/LogBarnesG.html}", note=[Accessed: 24-June-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_logbarnesg, organization={Wolfram Research}, title={LogBarnesG}, year={2022}, url={https://reference.wolfram.com/language/ref/LogBarnesG.html}, note=[Accessed: 24-June-2024 ]}