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]
Let us suppose that the Taylor series of reads as
Then the coefficients in the Kapteyn expansion of can be determined as follows. [4]: 571
The Kapteyn series of the powers of are found by Kapteyn himself: [1]: 103, [4]: 565
For it follows (see also [4]: 567 )
and for [4]: 566
Furthermore, inside the region , [4]: 559
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]
Let us suppose that the Taylor series of reads as
Then the coefficients in the Kapteyn expansion of can be determined as follows. [4]: 571
The Kapteyn series of the powers of are found by Kapteyn himself: [1]: 103, [4]: 565
For it follows (see also [4]: 567 )
and for [4]: 566
Furthermore, inside the region , [4]: 559