Kapteyn_series Knowpia

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 $f$ 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:^[2]^[3]

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 edit

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.^[4]^: 571

{\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 edit

The Kapteyn series of the powers of $z$ are found by Kapteyn himself:^[1]^: 103,^[4]^: 565

\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 ^[4]^: 567)

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

and for $n=2$ ^[4]^: 566

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

Furthermore, inside the region $D_{1}$ ,^[4]^: 559

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

References edit

^ ^a ^b Kapteyn, W. (1893). Recherches sur les functions de Fourier-Bessel. Ann. Sci. de l’École Norm. Sup., 3, 91-120.
^ ^a ^b Baricz, Árpád; Jankov Maširević, Dragana; Pogány, Tibor K. (2017). "Series of Bessel and Kummer-Type Functions". Lecture Notes in Mathematics. Cham: Springer International Publishing. doi:10.1007/978-3-319-74350-9. ISBN 978-3-319-74349-3. ISSN 0075-8434.
^ Borghi, Riccardo (2021). "Solving Kepler's equation via nonlinear sequence transformations". arXiv:2112.15154 [math.CA].
^ ^a ^b ^c ^d ^e Watson, G. N. (2011-06-06). A treatise on the theory of Bessel functions (1944 ed.). Cambridge University Press. OL 22965724M.