Barnes G-function

In mathematics, the Barnes G-function ${\displaystyle G(z)}$ is a function that is an extension of superfactorials to the complex numbers. It is related to the gamma function, the K-function and the Glaisher–Kinkelin constant, and was named after mathematician Ernest William Barnes. It can be written in terms of the double gamma function.

Formally, the Barnes G-function is defined in the following Weierstrass product form:

${\displaystyle G(1+z)=(2\pi )^{z/2}\exp \left(-{\frac {z+z^{2}(1+\gamma )}{2}}\right)\,\prod _{k=1}^{\infty }\left\{\left(1+{\frac {z}{k}}\right)^{k}\exp \left({\frac {z^{2}}{2k}}-z\right)\right\}}$

where ${\displaystyle \,\gamma }$ is the Euler–Mascheroni constant, exp(x) = e^x is the exponential function, and ${\displaystyle \Pi }$ denotes multiplication (capital pi notation).

The integral representation, which may be deduced from the relation to the double gamma function, is

${\displaystyle \log G(1+z)={\frac {z}{2}}\log(2\pi )+\int _{0}^{\infty }{\frac {dt}{t}}\left[{\frac {1-e^{-zt}}{4\sinh ^{2}{\frac {t}{2}}}}+{\frac {z^{2}}{2}}e^{-t}-{\frac {z}{t}}\right]}$

As an entire function, ${\displaystyle G}$ is of order two, and of infinite type. This can be deduced from the asymptotic expansion given below.