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

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

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

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

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:

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

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

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

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

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

${\begin{aligned}\alpha _{0}&=a_{0},\\\alpha _{n}&={\frac {1}{4}}\sum _{k=0}^{\left\lfloor {\frac {n}{2}}\right\rfloor }{\frac {(n-2k)^{2}(n-k-1)!}{k!(n/2)^{(n-2k+1)}}}a_{n-2k}\quad (n\geq 1).\end{aligned}}$

Examples

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

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

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 $D_{1}$,

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

See also

• Schlömilch's series