1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

326 CHAPTER 8 • REsIDUE THEORY

Keeping in mind the branch of logarithm that we're using, we then have

fct(z)dz= 1: f(x)dx+ 1-c. f(z)dz+ 1R f(x)dx+ lR f(z)dz

= 1 -rlnl:l +!n dx+j f(z)dz

• R X +a -C.

+1R ;nx 2 dx+ { f(z)dz

r X + a lei<

nlna .71"^2

= --+i -.

a 2a

If R^2 > a^2 , then by the ML inequality (Theorem 6.3)

I

r I (z)dzl = l {" lnR+iO iRei9d8 1

JcR Jo R2e•29 + a2

< R(lnR+n)7r

• R2 - a2 '

(8- 33 )

and L'Hopital's rule yields lim f.c f (z) dz= O. A similar computation shows
R-oo R
that lim J. f (z) dz = O. We use these results when we take corresponding
r-o+ c,,.
limits in Equations (8-33) to get

P.V. (Jo lnlxl+in dx+ [

00

lnx dx) = nlna +i7r

2

• 
  


• ~+~ k ~+~ a ~

Equating the real parts in this equation gives

l

oo lnx nlna
P.V. 2 2 dx = --.
0 x +a 2a

Remark 8.3 The theory of this section is not purely esoteric. Many applica-
tions of contour integrals surface in government and industry worldwide. Many
years ago, for example, a briefing was given at the Korean Institute for Defense
Analysis (KIDA) in which a sophisticated problem was analY2ed by means of a
contour integral whose path of integration was virtually identical to that given
in Figure 8.8. •

-------... EXERCISES FOR SECTION 8.6

Use residues to compute

roo dx

1. P.V. Jo!.
   x (1 + x)

roo dx

2. P.V. Jo ~.
   x (1 + x)