Error functions, Mordell integrals and an integral analogue of partial theta function

TL;DR

This paper introduces new transformations involving error functions and integrals related to theta functions, leading to exact, approximate, and asymptotic evaluations of complex integrals, including those involving the Riemann Xi-function.

Contribution

It presents novel transformations connecting error functions and theta function analogues, expanding methods for evaluating complex integrals and deriving asymptotic expansions.

Findings

Derived new error function transformations linked to theta functions.

Provided exact and approximate evaluations of non-elementary integrals.

Established asymptotic expansions involving the Riemann Xi-function.

Abstract

A new transformation involving the error function $\textup{erf}(z)$, the imaginary error function $\textup{erfi}(z)$, and an integral analogue of a partial theta function is given along with its character analogues. Another complementary error function transformation is also obtained which when combined with the first explains a transformation in Ramanujan's Lost Notebook termed by Berndt and Xu as the one for an integral analogue of theta functions. These transformations are used to obtain a variety of exact and approximate evaluations of some non-elementary integrals involving hypergeometric functions. Several asymptotic expansions, including the one for a non-elementary integral involving a product of the Riemann $\Xi$-function of two different arguments, are obtained, which generalize known results due to Berndt and Evans, and Oloa.