A Survey of Classical Mock Theta Functions

Chapter
Part of the Developments in Mathematics book series (DEVM, volume 23)

Abstract

In his last letter to Hardy, Ramanujan defined 17 functions M(q), | q | < 1, which he called mock θ-functions. He observed that as q radially approaches any root of unity ζ at which M(q) has an exponential singularity, there is a θ-function T ζ(q) with \(M(q) - {T}_{\zeta }(q) = O(1)\). Since then, other functions have been found which possess this property. We list various linear relations between these functions and develop their transformation laws under the modular group. We show that each mock θ-function is related to a member of a universal family (mock θ-conjectures). In recent years the subject has received new impetus and importance through a strong connection with the theory of Maass forms. The final section of this survey provides some brief remarks concerning these new developments.

Keywords

Mock theta functions q-series Modular forms 

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaLos AngelesUSA
  2. 2.Department of Mathematics and StatisticsUniversity of ReginaReginaCanada

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