Advanced Calculus fi..

&-4A-dvanced Calculus, Fifth EditionIn order to apply Theorem 33 it is necessary to have simple criteria to gdarafiteethat (8.90) holds. We list two such criteria here:I. If f (z) is rational and has a zero of order greater than 1 at oo, then (8.90)holds.For when )zl is sufficiently large,so that zf (z) has a zero at infinity. Now, by (8.1 l),I ln f(~e")Re'@d~I9 Inso that the integral must converge to 0 as R -+ m.Izf (z)I do,11. If g(z) is rational and has a zero of order 1 or greater at oo, then (8.90) holdsfor f (z) = em' 0.1 For a proof and further criteria we refer to page 115 of the treatise of Whittakerand Watson listed at the end of the chapter. Rule I1 makes possible the evaluation ofthe Fourier integral (Section 7.18),LO)wg(x)em'x dx = [_g(x) cos mx dx + i,-provided that both real integrals exist.EXAMPLE 3 The integralsx cos xx sin xdx and Srn - dx-, X2 + 1both exist by the corollary to Theorem 51 of Section 6.22. Hence .wTaking real and imaginary parts, we findx cos x x sinx xdx = -.eSince the first integral is an integral of an odd function, the value of 0 could bepredicted.•PROBLEMS1. Evaluate the following integrals: