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