17 Fourier, Laplace, and Mellin Transforms 17.1–17.4 Integral Transforms 17.11 Laplace t ransfo rm The Laplace transform of the function f (x), denoted by F (s), is defined by the integral F (s) = ∞ 0 f (x)e −sx dx, Re s > 0. The functions f (x) and F (s) are called a Laplace transform pair, and knowledge of either one enables the other to be recovered. If f is summable over all finite intervals, and there is a constant c for which ∞ 0 |f (x)|e −c|x| dx is finite, then the Laplace transform exists when s = σ + iτ is such that σ ≥ c . Setting F (s) = L [f (x); s] to emphasize the nature of the transform, we have the symbolic inverse result f (x) = L −1 [F (s); x] . The inversi on of the Laplace transform is accomplished for analytic functio ns F (s) of order O s −k with k > 1 by means of the inversion integral f (x) = 1 2πi γ +i∞ γ −i∞ F (s)e sx ds, where γ is a real constant that exceeds the real part of all the singularities of F (s). SN 30 17.12 Basic p ropert ies of the Laplace trans for m 1. 8 F or a and b arbitrary constants, L [af (x) + bg(x)] = aF (s) + bG(s) (line arity) 2. If n > 0 is a n int ege r and lim x→∞ f (x)e −sx = 0, then for x > 0, L f (n) (x); s = s n F (s) − s n−1 f (0) − s n−2 f (1) (0) − ··· − f (n−1) (0) (tr ansform of a derivative ) SN 32 1107
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