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.^[1][2] Let ${\displaystyle f}$ be a function analytic on the domain

${\displaystyle D_{a}=\left\{z\in \mathbb {C}$ :\Omega (z)=\left|{\frac {z\exp {\sqrt {1-z^{2}}}}{1+{\sqrt {1-z^{2}}}}}\right|\leq a\right\}}

with ${\displaystyle a<1}$. Then ${\displaystyle f}$ can be expanded in the form

${\displaystyle f(z)=\alpha _{0}+2\sum _{n=1}^{\infty }\alpha _{n}J_{n}(nz)\quad (z\in D_{a}),}$

where

${\displaystyle \alpha _{n}={\frac {1}{2\pi i}}\oint \Theta _{n}(z)f(z)dz.}$

The path of the integration is the boundary of ${\displaystyle D_{a}}$. Here ${\displaystyle \Theta _{0}(z)=1/z}$, and for ${\displaystyle n>0}$, ${\displaystyle \Theta _{n}(z)}$ is defined by

${\displaystyle \Theta _{n}(z)={\frac {1}{4}}\sum _{k=0}^{\left[{\frac {n}{2}}\right]}{\frac {(n-2k)^{2}(n-k-1)!}{k!}}\left({\frac {nz}{2}}\right)^{2k-n}}$

Kapteyn's series are important in physical problems. Among other applications, the solution ${\displaystyle E}$ of Kepler's equation ${\displaystyle M=E-e\sin E}$ can be expressed via a Kapteyn series:^[2][3]

${\displaystyle E=M+2\sum _{n=1}^{\infty }{\frac {\sin(nM)}{n}}J_{n}(ne).}$