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A new transformation involving the error function $\textup{erf}(z)$, theimaginary error function $\textup{erfi}(z)$, and an integral analogue of apartial theta function is given along with its character analogues. Anothercomplementary error function transformation is also obtained which whencombined with the first explains a transformation in Ramanujan's Lost Notebooktermed 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 approximateevaluations of some non-elementary integrals involving hypergeometricfunctions. Several asymptotic expansions, including the one for anon-elementary integral involving a product of the Riemann $\Xi$-function oftwo different arguments, are obtained, which generalize known results due toBerndt and Evans, and Oloa.

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