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 InterpretationBox[logG, LogBarnesG, Editable -> False, Selectable -> False](z)=(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.

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:

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  (2)

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

LogBarnesG is the sum of LogGamma functions:

Introduced in 2008
 (7.0)