BarnesG

BarnesG[z]

gives the Barnes G-function TemplateBox[{z}, BarnesG].

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 TemplateBox[{n}, BarnesG]=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 the functional equation TemplateBox[{{z, +, 1}}, BarnesG]=TemplateBox[{z}, BarnesG] TemplateBox[{z}, Gamma].
  • BarnesG[z] is an entire function of z with no branch cut discontinuities.
  • For certain special arguments, BarnesG automatically evaluates to exact values.
  • BarnesG can be evaluated to arbitrary numerical precision.
  • BarnesG automatically threads over lists.
  • BarnesG can be used with Interval and CenteredInterval objects. »

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  (26)

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:

BarnesG can be used with Interval and CenteredInterval objects:

Specific Values  (4)

Value at infinity:

Value at zero:

Evaluate symbolically at halfinteger arguments:

Evaluate symbolically at integer multiples of 1/4:

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 derivative 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 :

Taylor expansion at a generic point:

Find the series expansion at Infinity:

Applications  (5)

Integer values of BarnesG are related to the superfactorial:

BarnesG may be generated by symbolic solvers:

Compute the number of bits needed to store a large integer:

Compare to the exact result:

For an odd prime, a generalization of Wilson's theorem states that TemplateBox[{{{{(, {p, -, 1}, )}, !!}, =, {G, (, {p, +, 1}, )}}, p}, Mod]. Verify for the first few odd primes:

Define a Cauchy matrix constructed from the first positive integers and an integer shift :

Show the Cauchy matrix for arbitrary :

The determinant of this Cauchy matrix can be expressed in terms of BarnesG. Verify with a specific value of for the first few cases:

Properties & Relations  (2)

BarnesG satisfies a differential equation:

FindSequenceFunction can recognize the BarnesG sequence:

Neat Examples  (3)

Determinants of Hankel matrices built out of Bell numbers:

Determinants of Hankel matrices built out of Euler numbers:

The determinant of the Hilbert matrix can be expressed in terms of the Barnes G-function:

Verify the formula for the first few cases:

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

Text

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

CMS

Wolfram Language. 2008. "BarnesG." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. 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

BibTeX

@misc{reference.wolfram_2023_barnesg, author="Wolfram Research", title="{BarnesG}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/BarnesG.html}", note=[Accessed: 19-March-2024 ]}

BibLaTeX

@online{reference.wolfram_2023_barnesg, organization={Wolfram Research}, title={BarnesG}, year={2022}, url={https://reference.wolfram.com/language/ref/BarnesG.html}, note=[Accessed: 19-March-2024 ]}