The Ramanujan Journal

, Volume 7, Issue 1–3, pp 193–222 | Cite as

Modular Transformations of Ramanujan's Fifth and Seventh Order Mock Theta Functions

  • Basil Gordon
  • Richard J. Mcintosh
Article

Abstract

In his last letter to Hardy, Ramanujan defined 17 functions F(q), where |q| < 1. He called them mock theta functions, because as q radially approaches any point eir (r rational), there is a theta function Fr(q) with F(q) − Fr(q) = O(1). In this paper we obtain the transformations of Ramanujan's fifth and seventh order mock theta functions under the modular group generators τ → τ + 1 and τ → −1/τ, where q = eπiτ. The transformation formulas are more complex than those of ordinary theta functions. A definition of the order of a mock theta function is also given.

mock theta function modular form Mordell integral 

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Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Basil Gordon
    • 1
  • Richard J. Mcintosh
    • 2
  1. 1.Department of MathematicsUniversity of CaliforniaLos Angeles
  2. 2.Department of Mathematics and StatisticsUniversity of ReginaReginaCanada

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