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

Volume 177 / 2017

Atul Dixit, Arindam Roy, Alexandru Zaharescu Acta Arithmetica 177 (2017), 1-37 MSC: Primary 11M06, 33B20; Secondary 33C15, 34E05. DOI: 10.4064/aa8207-5-2016 Published online: 1 December 2016

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.

Authors

  • Atul DixitDepartment of Mathematics
    Tulane University
    New Orleans, LA 70118, U.S.A.
    E-mail: adixit@tulane.edu
    Current address:
    Department of Mathematics
    Indian Institute of Technology Gandhinagar
    Palaj, Gandhinagar 382355, India
    e-mail
  • Arindam RoyDepartment 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.
    e-mail
    e-mail
  • Alexandru ZaharescuDepartment 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
    e-mail

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