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Integrals of K and E from lattice sums

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Abstract

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.

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Acknowledgments

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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Authors and Affiliations

  1. Engineering Systems and Design, Singapore University of Technology and Design, 8 Somapah Rd, Singapore, 487372, Singapore

    J. G. Wan

  2. Department of Physics, Kings College London, Strand, London, WC2R 2LS, UK

    I. J. Zucker

Authors
  1. J. G. Wan
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  2. I. J. Zucker
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Correspondence to J. G. Wan.

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Wan, J.G., Zucker, I.J. Integrals of K and E from lattice sums. Ramanujan J 40, 257–278 (2016). https://doi.org/10.1007/s11139-015-9710-z

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  • DOI: https://doi.org/10.1007/s11139-015-9710-z

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