1550251515-Classical_Complex_Analysis__Gonzalez_

690 Chapter^9

_L 1_b4~-·b_3-=-t-·~·b_2~~--,:-~ •b1

-R 0

Fig. 9.la

Since f(z) is even, we have J~n J(x) dx = f 0 R f(x) dx and (9.11-2) becomes

R m

2 f f(x)dx+ f J(z)dz=21f"iLResf(z)

lo lr+ k=l z=bk

In view of (9.11-1) and Lemma 9.3, we have

R->oo lim jJ(z)dz =^0

r+

Hence if we take limits in (9.11-3) as R --+ oo we get

after dropping the factor 2.

(9.11-3)

(9.11-4)

Formula (9.11-4) applies, in particular, to rational functions of the form

J(x) = P(x)/Q(x), where P and Q are polynomials prime to each other

and such that the degree of Q is at least two units greater than the degree

of P, provided f is even and there are no poles of f lying on the real axis.

Clearly, if the degrees of P and Q satisfy the condition stipulated above,

we have f(z) = 0(1/zOI) with a ~ 2.

Example To evaluate J 0

00

x^4 dx/(x^6 + 1). The integrand admits the ex-

tension J(z) = z^4 /(z^6 +1) into the complex domain and f(z) satisfies the

conditions required for the application of (9.11-4), so that all we need is to

evaluate the right-hand side of that formula. The poles off lying in the

upper half-plane are bk = ei(rr/^6 +krr/^3 ) (k = 0, 1, 2, ), and by (9.8-2),

Res~= bts = ~b"kl

z=bk z^6 + 1 6bk 6