Open Access

The Mordell integral, quantum modular forms, and mock Jacobi forms

Research in Number Theory20151:1

https://doi.org/10.1007/s40993-015-0002-x

Received: 2 October 2014

Accepted: 2 October 2014

Published: 1 July 2015

Abstract

It is explained how the Mordell integral
$$\int_{\mathbb R} \frac{e^{\pi i \tau x^{2} - 2\pi zx}}{\cosh(\pi x)} dx $$

unifies the mock theta functions, partial (or false) theta functions, and some of Zagier’s quantum modular forms. As an application, we exploit the connections between q-hypergeometric series and mock and partial theta functions to obtain finite evaluations of the Mordell integral for rational choices of τ and z.

Mathematics Subject Classification: 11P55, 05A17

Keywords

Jacobi forms mock Jacobi Form partial Jacobi theta function False theta function Partial theta function Mock theta function Mordell integral Ramanujan