1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

8. 7 • THE ARGUMENT PRINCIPLE AND ROUCHE'S THEOREM 327

oo x' dx

3 .P.V.fo 2·

(1 + x)

l

oo X'l dx

4. P.V. fo -1 2 ·
   
   
   • x
     p roo In (x^2 + 1) d.x
     5 .. v.Jo x2+ 1 Hint: Use the integrand f (z) =
   
   
   
   

(^10) ~~~ii).
oo lnx dx
6 .P.V.fo 2·
(1 + x^2 )

7. P.V. J 000 In~~:.. x), where 0 <a< 1.

8 P V

roo lnx d.x

... Jo 2 , where a > O.
(x+ a)

J

oo sinx.. exp(iz)

9. P.V. _ 00 x dx. Hint: Use the mtegrand f (z) = z
   contour C in Figure 8.8. Let r--+ 0 and R--+ oo.

and the

oo sin^2 x.. 1 - exp (i2z)

IO. P.V. J_ 00 - 2 - dx. Hint: Use the mtegrand f (z) = 2 and

x z

the contour C in Figure 8.8. Let r --+ 0 and R --+ oo.

11. The Fresnel integrals J 0
    00

cos (x^2 ) dx and f 0

00

sin (x^2 ) dx are important in

the study of optics. Use the integrand f (z) =exp (- z^2 ) and the contour C

shown in Figure 8.9, and let R --+ oo to get the value of these integrals. Use

the fact from calculus that J;" e-x• d.x = Jl

y

R

Figure 8.9

8.7 The Argument Principle and Rouche's Theorem

We now derive two results based on Cauchy's residue t heorem. They have im-

portant practical applications and pertain only to functions all of whose isolated

singularities are poles.