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
be a function analytic on the domain
![{\displaystyle 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\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cfc113eb465cb059003ce89c75a8ae910f75a9d7)
with
. Then
can be expanded in the form
![{\displaystyle f(z)=\alpha _{0}+2\sum _{n=1}^{\infty }\alpha _{n}J_{n}(nz)\quad (z\in D_{a}),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6b8da1f27b466e55243bc15bb11447d72255862)
where
![{\displaystyle \alpha _{n}={\frac {1}{2\pi i}}\oint \Theta _{n}(z)f(z)dz.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6dd4d54b36c20a440210d5778bde5d8ef19ae19c)
The path of the integration is the boundary of
. Here
, and for
,
is defined by
![{\displaystyle \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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ef205d0047b130d680043e9e60c40da27a5e6ef)
Kapteyn's series are important in physical problems. Among other applications, the solution
of Kepler's equation
can be expressed via a Kapteyn series:[2][3]
![{\displaystyle E=M+2\sum _{n=1}^{\infty }{\frac {\sin(nM)}{n}}J_{n}(ne).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5edac1b630923a8e8440c3292afed643167ab9f2)
Relation between the Taylor coefficients and the αn coefficients of a function
Let us suppose that the Taylor series of
reads as
![{\displaystyle f(z)=\sum _{n=0}^{\infty }a_{n}z^{n}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9764eb8794a27e4a9bb1a99df7a58cb94a3fb75)
Then the
coefficients in the Kapteyn expansion of
can be determined as follows.[4]: 571
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a0326452462ef6844d69acb3049a3c343fbcb81)
Examples
The Kapteyn series of the powers of
are found by Kapteyn himself:[1]: 103, [4]: 565
![{\displaystyle \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}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bd4e3d80706d0250167ee4db60c14c938e96100)
For
it follows (see also [4]: 567 )
![{\displaystyle z=2\sum _{k=0}^{\infty }{\frac {J_{2k+1}((2k+1)z)}{(2k+1)^{2}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5d2187777d1d13b4a071516c5868b4bab6a6232)
and for
[4]: 566
![{\displaystyle z^{2}=2\sum _{k=1}^{\infty }{\frac {J_{2k}(2kz)}{k^{2}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98cc2487f347516bbab11645805e1d12add978e5)
Furthermore, inside the region
,[4]: 559
![{\displaystyle {\frac {1}{1-z}}=1+2\sum _{k=1}^{\infty }J_{k}(kz).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f06af901761aeebe18ff5c53157b2b7de6bd379c)
See also
References
- ^ 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.