1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

8.4 • IMPROPER INTEGRALS INVOLVING TRIGONOMETR.!C FUNCTIONS 313

the residues with the aid of L'Hopital's rule:

R 1 / 1

.

1 1

. (z-1-i)exp(iz)
es , + i = 1m

• 
  
  
  
  • I+! z^4 + 4
  
  
  
  

Similarly,

= lim [l +i(z- 1 -i)]exp(iz)

Z-"J+i 4z3

exp(-l+i)

= 4(1+i)^3

sin 1 -cos 1 - i (cos 1 +sin1)

16e

R If

_

1

.

1

= cos 1 - sin 1 - i (cos 1+sin1)

es , +i

6

.

1 e

Using Equation (8-13), we get

1

00

co:x ~x = -27T [Im (Res[/, 1 + i]) +Im (Res[/ , - 1 +ii)]

• oo x +
  _ 7r(cosl +sinl)
  
  
  • 4e
  
  
  
  

We are almost ready to give the proof of Theorem 8.4, but first we need one
preliminary result.

t Lemma 8.1 (Jordan's lemma) Suppose that P and Qare polynomials of de-
gree m and n, respectively, where n ~ m + 1. If Cn is the upper semicircle
z = Rei^8 , for 0 ~ 8 ~ 1T, then

lim 1 exp (iz) p (z) dz= O.

R -oo CR Q(z)

P r oof: From n :'.'.:: m + 1, it follows that I G!~l I -+ 0 as lz l --+ oo. Therefore, for

any E > 0, there exists R. > 0 such that

I

P(z) 1 <~

Q (z) 1T

(8- 1 5)

whenever lzl ~ R •. Using the ML inequality {Theorem 6.3) together with In-
equality (8-15), we get

I

{ exp(iz)P(z) dzl ~ { ~ lei•i ldzl ,

Jen Q (z) JcR. 1T

(8-16)