The Ramanujan Journal

, Volume 40, Issue 2, pp 257–278 | Cite as

Integrals of K and E from lattice sums

  • J. G. Wan
  • I. J. Zucker


We give closed form evaluations for many families of integrals, whose integrands contain algebraic functions of the complete elliptic integrals K and E. Our methods exploit the rich structures connecting complete elliptic integrals, Jacobi theta functions, lattice sums, and Eisenstein series. Various examples are given, and along the way new (including 10-dimensional) lattice sum evaluations are produced.


Complete elliptic integral Lattice sum Jacobi theta function Dirichlet L-series Gamma function Hypergeometric series Mellin transform 

Mathematics Subject Classification

11M06 33C75 33E05 11F03 33C20 



We would like to thank Jon Borwein, Heng Huat Chan and Armin Straub for useful discussions. We are also very grateful to the anonymous referee for their helpful comments.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Engineering Systems and DesignSingapore University of Technology and DesignSingaporeSingapore
  2. 2.Department of PhysicsKings College LondonLondonUK

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