Green’s function for the one-dimensional Helmholtz equation: closed-form solution from its Fourier sine series

It is presented a way to obtain the closed form for Green’s function related to the nonhomogeneous one-dimensional Helmholtz equation with homogeneous Dirichlet conditions on the boundary of the domain from its Fourier sine series representation. A closed form for the sum of the series ${\sum }_{k=1}^{\mathrm{\infty }}\sin kx\sin ky/(k^2-{\alpha }^2)$ is found in the process.

Keywords
Green’s function method; Nonhomogeneous Helmholtz equation; Homogeneous Dirichlet conditions.