  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.
