17 Fourier, Laplace, and Mellin Transforms 17.1–17.4 Integral Transforms 17.11 Laplace t ransfo rm The Laplace transformof the functionf(x), denoted byF(s), is defined by the integral F(s) = ∞ 0 f(x)e −sx dx, Re s >0. The functionsf(x) and F(s) are called a Laplace transform pair, and knowledge of either one enables the other to be recovered. Iffis 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 whens = 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 ofF(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