Advances in Mathematical Physics
Volume 2009 (2009), Article ID 425164, 34 pages
Review Article

A Review of Procedures for Summing Kapteyn Series in Mathematical Physics

1Astronomical Institute, Universiteit Utrecht, Princetonplein 5, NL-3584CC Utrecht, The Netherlands
2Institut für Geowissenschaften, Naturwissenschaftliche Fakultät III, Martin-Luther-Universität Halle, 06099 Halle, Germany

Received 8 July 2009; Revised 4 November 2009; Accepted 4 December 2009

Academic Editor: M. Lakshmanan

Copyright © 2009 R. C. Tautz and I. Lerche. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Since the discussion of Kapteyn series occurrences in astronomical problems the wealth of mathematical physics problems in which such series play dominant roles has burgeoned massively. One of the major concerns is the ability to sum such series in closed form so that one can better understand the structural and functional behavior of the basic physics problems. The purpose of this review article is to present some of the recent methods for providing such series in closed form with applications to: (i) the summation of Kapteyn series for radiation from pulsars; (ii) the summation of other Kapteyn series in radiation problems; (iii) Kapteyn series arising in terahertz sideband spectra of quantum systems modulated by an alternating electromagnetic field; and (iv) some plasma problems involving sums of Bessel functions and their closed form summation using variations of the techniques developed for Kapteyn series. In addition, a short review is given of some other Kapteyn series to illustrate the ongoing deep interest and involvement of scientists in such problems and to provide further techniques for attempting to sum divers Kapteyn series.