The Ramanujan Journal

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

Integrals of K and E from lattice sums



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.


  1. 1.
    Andrews, G.E., Askey, R., Roy, R.: Special Functions. Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (2001)Google Scholar
  2. 2.
    Bailey, D.H., Borwein, J.M., Broadhurst, D.J., Glasser, M.L.: Elliptic integral evaluations of Bessel moments and applications. J. Phys. A. 41, 5203–5231 (2008)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Borwein, J.M., Borwein, P.B.: Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity. Wiley, New York (1987)MATHGoogle Scholar
  4. 4.
    Borwein, J.M., Glasser, M.L., McPhedran, R.C., Wan, J.G., Zucker, I.J.: Lattice Sums Then and Now. Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (2013)CrossRefMATHGoogle Scholar
  5. 5.
    Borwein, J.M., Straub, A., Wan, J.G.: Three-step and four-step random walk integrals. Exp. Math. 22, 1–14 (2013)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Hirschhorn, M.D.: A simple proof of an identity of Ramanujan. J. Aust. Math. Soc. Ser. A 34, 31–35 (1983)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Liouville, J.: Nombre des représentations d’un entier quelconque sous la forme d’une somme de dix carrés. J. Math. Pure. Appl. Sér. 11, 1–8 (1866)MathSciNetGoogle Scholar
  8. 8.
    Papanikolas, M.A., Rogers, M., Samart, D.: The Mahler measure of a Calabi–Yau threefold and special \(L\)-values. Math. Z. 276(3–4), 1151–1163 (2014)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Rogers, M.: Identities for the Ramanujan zeta function. Adv. Appl. Math. 51(2), 266–275 (2013)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Rogers, M., Wan, J.G., Zucker, I.J.: Moments of elliptic integrals and critical \(L\)-values. Ramanujan J. 5, 1–18 (2014)MathSciNetMATHGoogle Scholar
  11. 11.
    Wan, J.G.: Moments of products of elliptic integrals. Adv. Appl. Math. 48, 121–141 (2012)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Zagier, D.: Introduction to modular forms. In: Waldschmidt, M. (ed.) From Number Theory to Physics, pp. 238–291. Springer, Heidelberg (1992)CrossRefGoogle Scholar
  13. 13.
    Zhou, Y.: Legendre functions, spherical rotations, and multiple elliptic integrals. Ramanujan J. 34, 373–428 (2014)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Zhou, Y.: On some integrals over the product of three Legendre functions. Ramanujan J. 35(2), 311–326 (2014)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Zucker, I.J.: Exact results for some lattice sums in 2, 4, 6 and 8 dimensions. J. Phys. A 7(13), 1568–1575 (1974)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Zucker, I.J.: \(70+\) Years of the Watson integrals. J. Stat. Phys. 145, 591–612 (2011)MathSciNetCrossRefMATHGoogle Scholar

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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