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 [math]\displaystyle{ f }[/math] be a function analytic on the domain

[math]\displaystyle{ 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\} }[/math]

with [math]\displaystyle{ a\lt 1 }[/math]. Then [math]\displaystyle{ f }[/math] can be expanded in the form

[math]\displaystyle{ f(z) = \alpha_0 + 2\sum_{n=1}^\infty \alpha_n J_n(nz)\quad(z\in D_a), }[/math]

where

[math]\displaystyle{ \alpha_n = \frac{1}{2\pi i}\oint\Theta_n(z)f(z)dz. }[/math]

The path of the integration is the boundary of [math]\displaystyle{ D_a }[/math]. Here [math]\displaystyle{ \Theta_0(z)=1/z }[/math], and for [math]\displaystyle{ n\gt 0 }[/math], [math]\displaystyle{ \Theta_n(z) }[/math] is defined by

[math]\displaystyle{ \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} }[/math]

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

[math]\displaystyle{ E=M+2\sum_{n=1}^\infty\frac{\sin(nM)}{n}J_n(ne). }[/math]

Relation between the Taylor coefficients and the [math]\displaystyle{ \alpha_n }[/math] coefficients of a function

Let us suppose that the Taylor series of [math]\displaystyle{ f }[/math] reads as

[math]\displaystyle{ f(z)=\sum_{n=0}^\infty a_nz^n. }[/math]

Then the [math]\displaystyle{ \alpha_n }[/math] coefficients in the Kapteyn expansion of [math]\displaystyle{ f }[/math] can be determined as follows.

[math]\displaystyle{ \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} }[/math]

Examples

The Kapteyn series of the powers of [math]\displaystyle{ z }[/math] are found by Kapteyn himself:

[math]\displaystyle{ \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). }[/math]

For [math]\displaystyle{ n = 1 }[/math] it follows (see also )

[math]\displaystyle{ z = 2 \sum_{k=0}^\infty \frac{J_{2k+1}((2k+1)z)}{(2k+1)^2}, }[/math]

and for [math]\displaystyle{ n = 2 }[/math]

[math]\displaystyle{ z^2 = 2 \sum_{k=1}^\infty \frac{J_{2k}(2kz)}{k^2}. }[/math]

Furthermore, inside the region [math]\displaystyle{ D_1 }[/math],

[math]\displaystyle{ \frac{1}{1-z} = 1 + 2 \sum_{k=1}^\infty J_k(kz). }[/math]

See also

• Schlömilch's series