A Generalization of the Complex Inversion Formula for the Laplace Transformation
P. G. Rooney · Proceedings of the American Mathematical Society · 1954
The purpose of this paper is to show that if /I 00 er" (t)dt, o then, under the same conditions as those for the complex inversion formula1 (see [4, chap.II, §7]), /> 7+100(5/)-Vi(l; 1 -/S; st)fis)ds, y-too if Re /8>0, /3=^ 1, 2, • • • .We shall also give conditions for the use of this formula when Re /3^0.The inversion formula II may be considered as a generalization of the complex inversion formula, since for |8 = 0, it reduces to that formula.The formula also holds, in a sense, when j3=l, 2, • • • ; for, since (r(i -0)-Wl; l -p-, st) = £ (r(r + l -ftyKsty, r=0 it seems natural to consider this expression, for /3 = w, equal to (st)ne" thus reproducing the complex inversion formula.Formula II can be derived under considerably more rigid conditions than those for the complex inversion formula, by means of integrals of fractional order.We shall not do this here.2. Re /3 positive.Theorem 1.If (1) e-y> it) £1(0, co), 7>0, /» 00 e-' it)dt, Re 5 ^ y, o (3) (u) is of bounded variation in a neighbourhood of u = t > 0, (4) Re 0 > 0, 0 * 1, 2, • • • ,