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 $$f$$ be a function analytic on the domain
 * $$D_a = \left\{z\in\mathbb{C}:\Omega(z)=\left|\frac{z\exp\sqrt{1-z^2}}{1+\sqrt{1-z^2}}\right|\le a\right\}$$

with $$a<1$$. Then $$f$$ can be expanded in the form
 * $$f(z) = \alpha_0 + 2\sum_{n=1}^\infty \alpha_n J_n(nz)\quad(z\in D_a),$$

where



\alpha_n = \frac{1}{2\pi i}\oint\Theta_n(z)f(z)dz. $$ The path of the integration is the boundary of $$D_a$$. Here $$\Theta_0(z)=1/z$$, and for $$n>0$$, $$\Theta_n(z)$$ is defined by



\Theta_n(z) = \frac14\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 $$E$$ of Kepler's equation $$M=E-e\sin E$$ can be expressed via a Kapteyn series:



E=M+2\sum_{n=1}^\infty\frac{\sin(nM)}{n}J_n(ne). $$

Relation between the Taylor coefficients and the $α_{n}$ coefficients of a function
Let us suppose that the Taylor series of $$f$$ reads as

f(z)=\sum_{n=0}^\infty a_nz^n. $$ Then the $$\alpha_n$$ coefficients in the Kapteyn expansion of $$f$$ can be determined as follows.



\begin{align} \alpha_0 &= a_0,\\ \alpha_n &= \frac14\sum_{k=0}^{\left\lfloor\frac{n}2 \right\rfloor}\frac{(n-2k)^2(n-k-1)!}{k!(n/2)^{(n-2k+1)}}a_{n-2k}\quad(n\ge1). \end{align} $$

Examples
The Kapteyn series of the powers of $$z$$ are found by Kapteyn himself:



\left(\frac{z}{2}\right)^{n}=n^{2} \sum_{m=0}^\infty \frac{(n+m-1)!}{(n+2 m)^{n+1}\, m!} J_{n+2 m}\{(n+2 m) z\}\quad(z\in D_1). $$

For $$n = 1$$ it follows (see also )

z = 2 \sum_{k=0}^\infty \frac{J_{2k+1}((2k+1)z)}{(2k+1)^2}, $$

and for $$n = 2$$

z^2 = 2 \sum_{k=1}^\infty \frac{J_{2k}(2kz)}{k^2}. $$

Furthermore, inside the region $$D_1$$,

\frac{1}{1-z} = 1 + 2 \sum_{k=1}^\infty J_k(kz). $$