On an integral representation of a class of Kapteyn (Fourier-Bessel) series: Kepler's equation, radiation problems and Meissel's expansion (Q993759)

An integral representation of a class of Kapteyn series \[ \gamma(a,b,c,d,g,s,\varepsilon,\nu,\varphi)=\sum_{n=1}^\infty \frac{a^n} {(dn+b)^s}J_{gn+\nu}[(gn+\nu)\varepsilon]\sin(2\pi nc+\varphi),\tag{1} \] where \(J_{n}(z)\) is the Bessel function of the first kind \(J_{n}(z)= \frac1\pi\int_0^\pi\cos[n\theta-z\sin\theta]\,d\theta\) is considered. An integral expression for the series in (1) is given in Theorem 1. Using this result the transcendental equation \(E-\varepsilon\sin E=M\) for \(E\in(0,\pi)\), \(M\in(0,\pi)\), and \(\varepsilon\in(0,1]\) is analyzed and an integral representation of the Kepler's solution \(E\) is obtained. The series arising in radiation problems are considered and its integral forms are found. For the series \(\sum_{n=1}^\infty \frac{J_{2n}(2n\varepsilon)}{n^{2m}}\) with \(\varepsilon\in(0,1]\) which is a polynomial in \(\varepsilon\), an integral representation is obtained.