PT - JOURNAL ARTICLE
AU - Asatryan, A. A.
TI - Summation of a Schlömilch type series
DP - 2015 Nov 08
TA - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science
VI - 471
IP - 2183
4099 - http://rspa.royalsocietypublishing.org/content/471/2183/20150359.short
4100 - http://rspa.royalsocietypublishing.org/content/471/2183/20150359.full
AB - The ability to accurately and efficiently characterize multiple scattering of waves of different nature attracts substantial interest in physics. The advent of photonic crystals has created additional impetus in this direction. An efficient approach in the study of multiple scattering originates from the Rayleigh method, which often requires the summation of conditionally converging series. Here summation formulae have been derived for conditionally convergent Schlömilch type series ∑s=−∞∞Zn(|sD−x|)×e−inarg(sD−x) eisDsinθ0, where Zn(z) stands for any of the following cylindrical functions of integer order: Bessel functions Jn(z), Neumann functions Y n(z) or Hankel functions of the first kind Hn(1)(z)=Jn(z)+iYn(z). These series arise in two-dimensional scattering problems on diffraction gratings with multiple inclusions per unit cell. It is shown that the Schlömilch series involving Hankel functions or Neumann functions can be expressed as an absolutely converging series of elementary functions and a finite sum of Lerch transcendent functions, while the Schlömilch series of Bessel functions can be transformed into a finite sum of elementary functions. The closed-form expressions for the Coates's integrals of integer order have also been found. The derived equations have been verified numerically and their accuracy and efficiency has been demonstrated.