Barnes G-function

Barnessche the function typically designated with is a function which represents an extension of the super faculties to the complex numbers. It is related to the gamma function, the function and the constants of Glaisher - Kinkelin and is named after the mathematician Ernest William Barnes.

Formally, the Barnessche function is defined in the form of a Weierstrass product as

Where γ denotes the Euler - Mascheroni constant.

Difference equation, functional equation and special values

The Barnessche function satisfies the difference equation

With the normalization implies the difference equation, the following values ​​for integer arguments assume:

So that

Where the gamma function and the function call. The difference equation defines the function clearly when the convexity condition is detected.

The difference equation of the function and the functional equation of the gamma function provide the following functional equation for the function, as originally demonstrated by Hermann Kinkelin:

Multiplication formula

As the gamma function satisfies the function multiplication formula:

Wherein a function is represented by

Is given. This is the derivative of the Riemann zeta - function and the constant of Glaisher - Kinkelin.

Asymptotic Expansion

The function has the following asymptotic expansion, which was found by Barnes:

Herein, the Bernoulli numbers and the constant of Glaisher - Kinkelin. ( Note that at the time of Barnes Bernoulli number was written as. This convention is no longer used. ) The development is valid for each sector, which does not contain the negative real axis.

Weblink

• Eric W. Weisstein: Barnes -Function. In: MathWorld (English).