Kapteyn series

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. 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:

Relation between the Taylor coefficients and the coefficients of a function

Let us suppose that the Taylor series of reads as
Then the coefficients in the Kapteyn expansion of can be determined as follows.

The Kapteyn series of the powers of are found by Kapteyn himself:
For it follows
and for
Furthermore, inside the region,