New applications of Poisson's summation formula

Abstract

Fourier expansions in modified Bessel functions are evaluated in terms of elementary functions. The results are obtained in the form of rapidly convergent series so that numerical computations are quite easy. This leads to the possibility of tabulating the modified Bessel functions in their critical domain. A physical application is also given: simple expansions giving the Madelung constant for cubic crystallographic structures are established; they involve only elementary functions and exhibit the greatest known accuracy.