  In this paper we present a spectral decomposition of solutions to
relativistic wave equations described on horizon penetrating hyperboloidal
slices within a given Schwarzschild-black-hole background. The wave equa- tion
in question is Laplace-transformed which leads to a spatial differential
equation with a complex parameter. For initial data which are analytic with
respect to a compactified spatial coordinate, this equation is treated with the
help of the Mathematica-package in terms of a sophisticated Taylor series
analysis. Thereby, all ingredients of the desired spectral decomposition arise
explicitly to arbitrarily prescribed accuracy, including quasi normal modes,
quasi normal mode amplitudes as well as the jump of the Laplace-transform along
the branch cut. Finally, all contributions are put together to obtain via the
inverse Laplace transformation the spectral de- composition in question. The
paper explains extensively this procedure and includes detailed discussions of
relevant aspects, such as the definition of quasi normal modes and the question
regarding the contribution of infinity frequencies modes to the early time
response of the black hole.
