LogBarnesG

LogBarnesG[z]

gives the logarithm of the Barnes G-function TemplateBox[{z}, LogBarnesG].

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 TemplateBox[{z}, LogBarnesG]=(z-1) (TemplateBox[{z}, LogGamma]-z/2)+1/2 z log(2 pi)-TemplateBox[{{-, 2}, z}, PolyGamma2].
  • 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

open allclose all

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  (28)

Numerical Evaluation  (6)

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:

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 LogBarnesG function using MatrixFunction:

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 TemplateBox[{z}, LogBarnesG]:

Plot the imaginary part of TemplateBox[{z}, LogBarnesG]:

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:

TraditionalForm formatting:

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 ^(th) 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: 05-October-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: 05-October-2024 ]}