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:

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

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

The Barnes G-function satisfies the functional equation

with normalization ${\displaystyle G(1)=1}$. Note the similarity between the functional equation of the Barnes G-function and that of the Euler gamma function:

The functional equation implies that ${\displaystyle G}$ takes the following values at integer arguments: