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

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

with $a<1$. Then $f$ can be expanded in the form

$f(z)={\alpha }_0+2\sum _{n=1}^{\mathrm{\infty }}{\alpha }_n{J}_n(nz)(z\in D_a),$

where

${\alpha }_n=\frac{1}{2\pi i}\oint {\mathrm{\Theta }}_n(z)f(z)dz.$

The path of the integration is the boundary of $D_a$. Here ${\mathrm{\Theta }}_0(z)=1/z$, and for $n>0$, ${\mathrm{\Theta }}_n(z)$ is defined by

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

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

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

Relation between the Taylor coefficients and the ${\alpha }_n$ coefficients of a function

Let us suppose that the Taylor series of $f$ reads as

$f(z)=\sum _{n=0}^{\mathrm{\infty }}{a}_n{z}^n.$

Then the ${\alpha }_n$ coefficients in the Kapteyn expansion of $f$ can be determined as follows.

$\begin{array}{rl}{\alpha }_0& =a_0,\\ {\alpha }_n& =\frac{1}{4}\sum _{k=0}^{\lfloor \frac{n}{2}\rfloor }\frac{(n-2k{)}^2(n-k-1)!}{k!(n/2{)}^{(n-2k+1)}}{a}_{n-2k}(n\ge 1).\end{array}$

Examples

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

${\left(\frac{z}{2}\right)}^n=n^2\sum _{m=0}^{\mathrm{\infty }}\frac{(n+m-1)!}{(n+2m{)}^{n+1}m!}{J}_{n+2m}\{(n+2m)z\}(z\in D_1).$

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

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

and for $n=2$

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

Furthermore, inside the region $D_1$,

$\frac{1}{1-z}=1+2\sum _{k=1}^{\mathrm{\infty }}{J}_k(kz).$

See also

• Schlömilch's series