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.
be a function analytic on the domain with
can be expanded in the form where The path of the integration is the boundary of
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] Let us suppose that the Taylor series of
coefficients in the Kapteyn expansion of
[4]: 571 The Kapteyn series of the powers of
are found by Kapteyn himself:[1]: 103, [4]: 565 For
[4]: 566 Furthermore, inside the region