# 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 for positive integers and is otherwise defined as .
• The Barnes G-function satisfies the functional equation .
• 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:

### 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 . 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: