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

Atul Dixit¹ ; Arindam Roy² ; Alexandru Zaharescu³

¹ Department of Mathematics Tulane University New Orleans, LA 70118, U.S.A. E-mail: [e-mail] Current address: Department of Mathematics Indian Institute of Technology Gandhinagar Palaj, Gandhinagar 382355, India
² Department of Mathematics University of Illinois 1409 West Green Street Urbana, IL 61801, U.S.A. and Department of Mathematics Rice University 6100 Main St. Houston, TX 77005, U.S.A.
³ Department of Mathematics University of Illinois 1409 West Green Street Urbana, IL 61801, U.S.A. and Simion Stoilow Institute of Mathematics of the Romanian Academy P.O. Box 1-764 RO-014700 Bucureşti, Romania

Acta Arithmetica, Tome 177 (2017) no. 1, pp. 1-37

Résumé

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.

Détail

DOI : 10.4064/aa8207-5-2016

Keywords: transformation involving error function operatorname erf imaginary error function operatorname erfi integral analogue partial theta function given along its character analogues another complementary error function transformation obtained which combined first explains transformation ramanujan lost notebook termed berndt integral analogue theta functions these transformations obtain variety exact approximate evaluations non elementary integrals involving hypergeometric functions several asymptotic expansions including non elementary integral involving product riemann xi function different arguments obtained which generalize known results due berndt evans oloa