A class of infinite sums and integrals

Shail, R.

doi:10.1090/S0025-5718-00-01211-4

A class of infinite sums and integrals
HTML articles powered by AMS MathViewer

by R. Shail;

Math. Comp. 70 (2001), 789-799

DOI: https://doi.org/10.1090/S0025-5718-00-01211-4

Published electronically: March 1, 2000

PDF | Request permission

Abstract:

In this paper closed-form sums are given for various slowly-convergent infinite series which arise essentially from the differentiation of Dirichlet $L$-series. Some associated integrations are also considered. A small number of the results appear in standard tables, but most seem to be new.

References

D. Borwein, J. M. Borwein, R. Shail, and I. J. Zucker, Energy of static electron lattices, J. Phys. A 21 (1988), no. 7, 1519–1531. MR 951042
R. Shail, Some logarithmic lattice sums, J. Phys. A 28 (1995), no. 23, 6999–7009. MR 1381155
Enrique A. González-Velasco, The homotopic proof of Cauchy’s integral theorem, Internat. J. Math. Ed. Sci. Tech. 11 (1980), no. 2, 189–191. MR 580971, DOI 10.1080/0020739800110207
K. Chandrasekharan, Introduction to analytic number theory, Die Grundlehren der mathematischen Wissenschaften, Band 148, Springer-Verlag New York, Inc., New York, 1968. MR 249348
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
I. J. Zucker and M. M. Robertson, Some properties of Dirichlet $L$-series, J. Phys. A 9 (1976), no. 8, 1207–1214. MR 412124
I. J. Zucker and M. M. Robertson, Exact values for some two-dimensional lattice sums, J. Phys. A 8 (1975), 874–881. MR 421516
E. T. Whittaker and G. N. Watson, A course of modern analysis. An introduction to the general theory of infinite processes and of analytic functions: with an account of the principal transcendental functions, Cambridge University Press, New York, 1962. Fourth edition. Reprinted. MR 178117
Nico M. Temme, Special functions, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1996. An introduction to the classical functions of mathematical physics. MR 1376370, DOI 10.1002/9781118032572