1550251515-Classical_Complex_Analysis__Gonzalez_

Singularities/Residues/ Applications

r+

-R R x

Fig. 9.19

By changing x into -x in the first integral above, we get

fr e-ix dx = -JR e-ix dx

jR X r X

and combining with the third integral, we have

J

R eix - e-ix 1R sin x

----dx = 2i --dx

r X r X

Thus (9.11-26) can be written as

Since

1

R sinx J eiz J eiz

2i --dx + - dz+ - dz = 0

r X Z Z

'Y- r+

eiz

Res-=1

z=O Z

and by Lemma 9.2,

lim j eiz dz = 0

R-+oo z

r+

703

(9.11-27)

by taking limits in (9.11-27) first as r -+ O, then as R-+ oo, we obtain

or

1

.^00 SlnX • ,
2i --dx - i71' = 0
0 x

{

00

sinx dx = ?!:.

} 0 x 2

(9.11-28)

the convergence of the integral being assured in view of (9.11-27).

Remarks From (9.11-28) other improper integrals can be derived by sim-

ple changes of variable. For instance, if we let x = mu, m > O, we