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arXiv · 1605.08904

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

Atul Dixit·Arindam Roy·Alexandru Zaharescu
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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 $Ξ$-function of two different arguments, are obtained, which generalize known results due to Berndt and Evans, and Oloa.

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Atul Dixit, Arindam Roy, Alexandru Zaharescu. 2016-05-28. Error functions, Mordell integrals and an integral analogue of partial theta function. https://arxiv.org/abs/1605.08904

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