Advanced Calculus fi..

, .rJii' Chapter 6 Infinite SeriesEXAMPLEThe integralis uniformly convergent for x > 1, sinceand the integral0 5 e-Xt2 5 e-'2 = M(t) for x s 1,exists, as a comparison with&. dreveals. Hence (6.74) defines a function F(x). Theorem 56 shows that F(x) is continuousfor x > 1. The integralis also uniformly convergent, sinceIfor x 2 1 and t sufficiently large. Hence"e-xr2 dt = - t'e-"2 dt, x 2 1.dx"hmAs in the preceding section, the theory extends to improper integrals over a finiteinterval without essential change.iAn integral from -oo to oo would normally be decomposed into integrals from -00to 0 and from 0 to oo, as in Section 4.1. If the last two integrals converge, the givenintegral converges; otherwise, it diverges. When this process leads to divergence, onemay be able to salvage the integral by the following procedure, which has importantapplications to Laplace and Fourier transforms.Let f be continuous for -oo < x < oo (or, more generally, let f be such that fhas an integral over each finite interval). Then the Cauchy principal value (briefly,principal value) of the integral of f from -oo to oo is the limitif this limit exists. When it does, one denotes the value by