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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 equation 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 quasinormal modes and quasinormal 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 decomposition in question. The paper explains extensively this procedure and includes detailed discussions of relevant aspects, such as the definition of quasinormal modes and the question regarding the contribution of infinity frequency modes to the early time response of the black hole.
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