Representations for the derivative at zero and finite parts of the Barnes zeta function

Abstract

We provide new representations for the finite parts at the poles and the derivative at zero of the Barnes zeta function in any dimension in the general case. These representations are in the forms of series and limits. We also give an integral representation for the finite parts at the poles. Similar results are derived for an associated function, which we term homogeneous Barnes zeta function. Our expressions immediately yield analogous representations for the logarithm of the Barnes Gamma function, including the particular case also known as multiple Gamma function.

This is the Author's Original Manuscript of an article published in Integral Transforms and Special Functions (Taylor & Francis). Published article available at http://www.tandfonline.com/eprint/ME3GywdpTKcevj9jUE5t/full (There are no differences in content between the versions.)

References in corpus (2)

• Finite-size effects on the Bose-Einstein condensation critical temperature in a harmonic trap
• Zeta Determinant for Laplace Operators on Riemann Caps

Cited by in corpus (1)

• Spectral determinant on Euclidean isosceles triangle envelopes of fixed area as a function of angles: absolute minimum and small-angle asymptotics