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 formula_1 be a function analytic on the domain
with formula_3. Then formula_1 can be expanded in the form
where
The path of the integration is the boundary of formula_7. Here formula_8, and for formula_9, formula_10 is defined by
Kapteyn's series are important in physical problems. Among other applications, the solution formula_12 of Kepler's equation formula_13 can be expressed via a Kapteyn series:
Relation between the Taylor coefficients and the formula_15 coefficients of a function.
Let us suppose that the Taylor series of formula_1 reads as
Then the formula_15 coefficients in the Kapteyn expansion of formula_1 can be determined as follows.
Examples.
The Kapteyn series of the powers of formula_21 are found by Kapteyn himself:
Furthermore, inside the region formula_23,