Error bounds and exponential improvement for the asymptotic expansion of the Barnes $G$-function

TL;DR

This paper develops new integral representations for the Barnes G-function's asymptotic expansion, providing explicit error bounds and revealing the smooth emergence of exponentially small terms along the imaginary axis, demonstrating a Berry transition.

Contribution

It introduces novel integral representations for the remainder term, enabling simpler error bounds and an exponentially improved expansion that confirms the Berry transition in the G-function's asymptotics.

Findings

Explicit, numerically computable error bounds are derived.

Exponentially small terms appear smoothly along the imaginary axis.

An exponentially improved asymptotic expansion confirms the Berry transition.

Abstract

In this paper we establish new integral representations for the remainder term of the known asymptotic expansion of the logarithm of the Barnes $G$-function. Using these representations, we obtain explicit and numerically computable error bounds for the asymptotic series, which are much simpler than the ones obtained earlier by other authors. We find that along the imaginary axis, suddenly infinitely many exponentially small terms appear in the asymptotic expansion of the Barnes $G$-function. Employing one of our representations for the remainder term, we derive an exponentially improved asymptotic expansion for the logarithm of the Barnes $G$-function, which shows that the appearance of these exponentially small terms is in fact smooth, thereby proving the Berry transition property of the asymptotic series of the $G$-function.