Integrals of K and E from lattice sums

被引：6

作者：

Wan, J. G. ^{[1
]}

Zucker, I. J. ^{[2
]}

机构：

[1] Singapore Univ Technol & Design, Engn Syst & Design, 8 Somapah Rd, Singapore 487372, Singapore

[2] Kings Coll London, Dept Phys, London WC2R 2LS, England

来源：

RAMANUJAN JOURNAL | 2016年 / 40卷 / 02期

关键词：

Complete elliptic integral; Lattice sum; Jacobi theta function; Dirichlet L-series; Gamma function; Hypergeometric series; Mellin transform; ELLIPTIC INTEGRALS; LEGENDRE FUNCTIONS; RAMANUJAN;

D O I：

10.1007/s11139-015-9710-z

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

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.

引用

页码：257 / 278

页数：22

共 16 条

[1]

[Anonymous], ENCY MATH ITS APPL

[2] Elliptic integral evaluations of Bessel moments and applications [J].

Bailey, David H. ;

Borwein, Jonathan M. ;

Broadhurst, David ;

Glasser, M. L. .

JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, 2008, 41 (20)

[3]

Borwein J., 1987, Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity

[4]

Borwein JM, 2013, ENCYCLOP MATH APPL, V150, P1, DOI 10.1017/CBO9781139626804

[5] Three-Step and Four-Step Random Walk Integrals [J].

Borwein, Jonathan M. ;

Straub, Armin ;

Wan, James .

EXPERIMENTAL MATHEMATICS, 2013, 22 (01) :1-14

[6] A SIMPLE PROOF OF AN IDENTITY OF RAMANUJAN [J].

HIRSCHHORN, MD .

JOURNAL OF THE AUSTRALIAN MATHEMATICAL SOCIETY SERIES A-PURE MATHEMATICS AND STATISTICS, 1983, 34 (FEB) :31-35

[7]

LIOUVILLE J, 1966, J MATH PURES APPL, V0011, P00001

[8]

Papanikolas MA, 2014, MATH Z, V276, P1151, DOI 10.1007/s00209-013-1238-6

[9]

Rogers M., 2014, RAMANUJAN J, V5, P1

[10] Identities for the Ramanujan zeta function [J].

Rogers, Mathew .

ADVANCES IN APPLIED MATHEMATICS, 2013, 51 (02) :266-275

← 1 2 →