BarnesG

BarnesG[z]

gives the Barnes G-function .

Details

  • BarnesG is also known as the double gamma function.
  • Mathematical function, suitable for both symbolic and numeric manipulation.
  • The Barnes G-function is defined as G(n)=product_(k=1)^(n-1)TemplateBox[{k}, Gamma] for positive integers and is otherwise defined as TemplateBox[{z}, BarnesG]=(2 pi)^(z/2) exp((z-1) (TemplateBox[{z}, LogGamma]-z/2)-TemplateBox[{{-, 2}, z}, PolyGamma2]).
  • The Barnes G-function satisfies .
  • BarnesG can be evaluated to arbitrary numerical precision.
  • BarnesG automatically threads over lists.

Examples

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Basic Examples  (5)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

Scope  (25)

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  (4)

Value at infinity:

Value at zero:

Evaluate symbolically at halfinteger orders:

Find the first positive maximum:

Visualization  (2)

Plot the BarnesG function:

Plot the real part of :

Plot the imaginary part of :

Function Properties  (10)

BarnesG is defined for all real and complex values:

Approximate function range of BarnesG:

BarnesG is an analytic function of x:

BarnesG is neither non-increasing nor non-decreasing:

BarnesG is not injective:

BarnesG is surjective:

BarnesG is neither non-negative nor non-positive:

BarnesG has no singularities or discontinuities:

BarnesG is neither convex nor concave:

TraditionalForm formatting:

Differentiation  (2)

First derivatives with respect to z:

Higher derivatives with respect to z:

Plot the higher derivatives with respect to z:

Series Expansions  (3)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Find the series expansion at Infinity:

Taylor expansion at a generic point:

Applications  (2)

Integral values of BarnesG are related to superfactorial:

BarnesG may be generated by symbolic solvers:

Properties & Relations  (2)

BarnesG satisfies a differential equation:

FindSequenceFunction can recognize the BarnesG sequence:

Neat Examples  (1)

Determinants of Hankel matrices built out of Bell numbers:

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

Text

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

BibTeX

@misc{reference.wolfram_2021_barnesg, author="Wolfram Research", title="{BarnesG}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/BarnesG.html}", note=[Accessed: 21-October-2021 ]}

BibLaTeX

@online{reference.wolfram_2021_barnesg, organization={Wolfram Research}, title={BarnesG}, year={2008}, url={https://reference.wolfram.com/language/ref/BarnesG.html}, note=[Accessed: 21-October-2021 ]}

CMS

Wolfram Language. 2008. "BarnesG." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/BarnesG.html.

APA

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