Definition and Elementary Properties

Abstract

Let f(t) be an arbitrary function defined on the interval 0 ≤ t < ∞; then

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaabm % aabaGaamiuaaGaayjkaiaawMcaaiabg2da9maapehabaGaamyzamaa % CaaaleqabaGaeyOeI0IaamiCaiaadshaaaGccaWGMbWaaeWaaeaaca % WG0baacaGLOaGaayzkaaGaamizaiaadshaaSqaaiaaicdaaeaacqGH % EisPa0Gaey4kIipaaaa!4802!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$F\left( P \right) = \int\limits_0^\infty {{e^{ - pt}}f\left( t \right)dt} $$

(1)

is the Laplace transform of f(t), provided that the integral exists. We shall confine our attention to functions f(t) which are absolutely integrable on any interval 0 ≤ t ≤ a, and for which F(α) exists for some real α. It may readily be shown that for such a function F(p) is an analytic function of p for Re(p) > α, as follows. First note that the functions

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dy2aae % WaaeaacaWGWbGaaiilaiaadsfaaiaawIcacaGLPaaacqGH9aqpdaWd % XbqaaiaadwgadaahaaWcbeqaaiabgkHiTiaadchacaWG0baaaOGaam % OzamaabmaabaGaamiDaaGaayjkaiaawMcaaiaadsgacaWG0baaleaa % caaIWaaabaGaamivaaqdcqGHRiI8aaaa!4A10!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\phi \left( {p,T} \right) = \int\limits_0^T {{e^{ - pt}}f\left( t \right)dt} $$

(2)

are analytic in p, and then that φ(p,T) converges uniformly to F(p) in any bounded region of the p plane satisfying Re(p) > α, as T → ∞. It follows from a standard theorem on uniform convergence² that F(p) is analytic in the half-plane Re(p) > α.

The results given in this section may be found in many places. We mention in particular DITKIN & PRUDNIKOV (1965), DOETSCH (1971), and WIDDER (1944).