Determination of the Original Function by Means of Series Expansion of the Image Function
Having established a function F(s) as a L-transform, the following technique suggests itself as a means of determining the corresponding original function: Expand F(s) into a series of image functions having known original functions and then return this series to the original space term by term. Certain hypotheses must necessarily be satisfied, for the indicated process involves the interchange of L-transformation and infinite summation. For the special case that F(s) may be represented by a partial fraction expansion, conditions taken from the theory of functions which guarantee the legality of the termwise inverse transformation have been presented in Chapter 26. In this Chapter we derive a theorem of strictly analytical character which proves extremely practical in applications. Its verification requires Lebesgue integration; thus, we introduce two lemmas from the Le-besgue theory.
KeywordsSeries Expansion Entire Function Original Function Image Function Dirichlet Series
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