Barnes G-function

In mathematics, the Barnes G-function 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. Up to elementary factors, it is a special case 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, and ∏ is capital pi notation.

The Barnes G-function satisfies the functional equation

with normalisation G(1) = 1. Note the similarity between the functional equation of the Barnes G-function and that of the Euler Gamma function:

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