Kapteyn Series for the Scalar Synchrotron Radiation

We consider a family of the Kapteyn series of 2nd kind, with power functions ( \(n^{\lambda}\) ) as coefficients. For the generic case \(\lambda=1\) we represent the sum as a single-variable integral and study it in detail. It allows presenting the asymptotic expansion of the integral in a consistent way. In particular, it allows computing the scalar radiation in hypothetic three-dimensional world.