Summation of Fourier series with parameter by Laplace transforms (Q1310365)

Let \(f\) be a function defined in the open interval \((0,\infty)\). Then the Laplace transform \(F\) of \(f\) is given by \(F(s)= \int^ \infty_ 0 f(t)\exp(-st)dt\), where \(s\) is a real or complex parameter independent of \(t\). In this paper, the author has used Laplace transform to obtain the sums of \(\sum^ \infty_{n=1} F(n)\exp(int)\), \(\sum^ \infty_{n=1} F(\sigma+ in)\sin nt\), and \(\sum^ \infty_{n=1} F(\sigma+ in)\cos nt\), where \(\sigma\) is any real parameter. One of his theorems reads as: \[ \sum^ \infty_{n=1} F(n)\sin nt= {\sin t\over 2} \int^ \infty_ 0 f(x) {dx\over \text{cosh }x-\cos t} \] and \[ \sum^ \infty_{n=1} F(n)\cos nt= {1\over 2} \int^ \infty_ 0 f(x)\left({\text{sinh }x\over\text{cosh }x-\cos t}- 1\right)dx. \] He has also applied his results to find the sum of the series.