Kapteyn series is a series expansion of analytic functions on a domain in terms of the Bessel function of the first kind. Kapteyn series are named after Willem Kapteyn, who first studied such series in 1893.[1][2] Let be a function analytic on the domain
with . Then can be expanded in the form
where
The path of the integration is the boundary of . Here , and for , is defined by
Kapteyn's series are important in physical problems. Among other applications, the solution of Kepler's equation can be expressed via a Kapteyn series:[2][3]
Relation between the Taylor coefficients and the coefficients of a function[edit]
^ abKapteyn, W. (1893). Recherches sur les functions de Fourier-Bessel. Ann. Sci. de l’École Norm. Sup., 3, 91-120.
^ abBaricz, Árpád; Jankov Maširević, Dragana; Pogány, Tibor K. (2017). "Series of Bessel and Kummer-Type Functions". Lecture Notes in Mathematics. Cham: Springer International Publishing. doi:10.1007/978-3-319-74350-9. ISBN978-3-319-74349-3. ISSN0075-8434.