Kapteyn series in high intensity Compton scattering

• Published in: Journal of physics. A, Mathematical and theoretical Vol. 43; no. 11; p. 115207
• Main Authors: Lerche, I, Tautz, R C
• Format: Journal Article
• Language: English
• Published: Bristol IOP Publishing 19.03.2010; IOP
• Subjects: Elastic scattering; Emission; Exact sciences and technology; Exact solutions; Mathematical analysis; Mathematical models; Physics; Reduction; Scattering; Series (mathematics); Active galaxies; Intensity; Active galaxy nuclei; Physical parameter; Galaxy nuclei; Jets; Compton effect; Integral representation
• ISSN: 1751-8121; 1751-8113; 1751-8121
• DOI: 10.1088/1751-8113/43/11/115207

In discussing the signatures available from high intensity Compton scattering, a problem of considerable interest in photon scattering from active galactic nuclei and their emission jets, Harvey et al (2009 Phys. Rev. A 79 063407) showed that the properties of some other than conventional Kapteyn series play fundamental roles in determining the spectral output both with respect to frequency and with respect to emission angle. While they were able to provide bounds to the required series in terms of known Kapteyn series that could be summed analytically in closed form, only numerical analysis could take their investigation further as they demonstrated. The purpose of this paper is to show that the many Kapteyn series involved in the scattering problem can all be reduced either to analytic form or to a single Kapteyn series that cannot be evaluated in closed form but for which an integral representation is available. This reduction is of considerable benefit in controlling the correctness and accuracy of numerical investigations; the reduction also provides significant insight into some basic procedures for summing such unconventional Kapteyn series as well as allowing a better understanding of the dependence of the scattering on the physical parameters involved than would be the case directly from the series.