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

### Volume 177 / 2017

#### Abstract

A new transformation involving the error function $\operatorname{erf} (z)$, the imaginary error function $\operatorname{erfi} (z)$, 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.