The Complex Inversion Formula for the Absolutely Converging Laplace Transformation. The Fourier Transformation
Hitherto, without exception, we determined the image function F(s) of some given original function f(t). Clearly, often one is faced with the inverse problem, that is to find the corresponding original function f(t) of some given function F(s), which is known to be a L-transform. A large number of so-called “inversion formulae” is available which solve this problem, each being applicable under specific hypotheses. For practical applications by far the most important is formula (1.10) which was mentioned early in the beginning of this book; it was then derived from formulae (1.5) and (1.6) which pertain to the Fourier integral, without explicit enumeration of the necessary hypotheses. We shall now present these.
KeywordsBounded Variation Laplace Transformation Inversion Formula Finite Interval Integral Converge
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