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 {\displaystyle f} be a function analytic on the domain
D a = { z ∈ C : Ω ( z ) = | z exp 1 − z 2 1 + 1 − z 2 | ≤ a } {\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\}}
with a < 1 {\displaystyle a<1} . Then f {\displaystyle f} can be expanded in the form
f ( z ) = α 0 + 2 ∑ n = 1 ∞ α n J n ( n z ) ( z ∈ D a ) , {\displaystyle f(z)=\alpha _{0}+2\sum _{n=1}^{\infty }\alpha _{n}J_{n}(nz)\quad (z\in D_{a}),}
where
α n = 1 2 π i ∮ Θ n ( z ) f ( z ) d z . {\displaystyle \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 {\displaystyle D_{a}} . Here Θ 0 ( z ) = 1 / z {\displaystyle \Theta _{0}(z)=1/z} , and for n > 0 {\displaystyle n>0} , Θ n ( z ) {\displaystyle \Theta _{n}(z)} is defined by
Θ n ( z ) = 1 4 ∑ k = 0 [ n 2 ] ( n − 2 k ) 2 ( n − k − 1 ) ! k ! ( n z 2 ) 2 k − n {\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}}
Kapteyn's series are important in physical problems. Among other applications, the solution E {\displaystyle E} of Kepler's equation M = E − e sin E {\displaystyle M=E-e\sin E} can be expressed via a Kapteyn series:
E = M + 2 ∑ n = 1 ∞ sin ( n M ) n J n ( n e ) . {\displaystyle E=M+2\sum _{n=1}^{\infty }{\frac {\sin(nM)}{n}}J_{n}(ne).}
Relation between the Taylor coefficients and the αn coefficients of a function Let us suppose that the Taylor series of f {\displaystyle f} reads as
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