1550251515-Classical_Complex_Analysis__Gonzalez_

Singularities/Residues/Applications

Exercises 9.3

1. Show that j ( 3 ~

2

~

2

1 ~~~ ~ 4 ) = ~ 7ri, c+: z = 3eit, O $ t $ 271".

c+

J

(3z^7 + 4) dz. + it

2. Show that zB _
   1
   = 6m, C : z = 2e , 0 $ t $ 271".
   c+

679

3 Sh h t

J

. ow t a sin z d '(. h^1. ) c+ it

z^3 +z^2 +z+^1 z = 7rZ sm - sin 1 , : z = 2e

c+

0 < t < 271".

4. Sh:w t~at j tan7rzdz = -4ni, c+: z = neit, O $ t $ 271", (n > 0).
   c+


5. Show that j .dhz {-
   4
   7l"i n odd c+: z = (n + 1/ 2 )7reit, o $ t $
   sm z O neven
   c+

271" (n > 0).

6. Show that = -^1 / 2 7ri, c-: z = e-it, 0 $ t $ 271".

J

ezdz

z^2 (z^2 - 4)

c-

7. Show that j
   c+

Z^2 + Z -2 d -^17 7l"Z. c+.

( z - 1 )( z - 4) z - 24 ' · z -^2 e ' it^0 < t <^2 71" ·

8. Let P(z) = aozn +···+an (ao =/: 0) and Q(z) = bozm + · · · + bm
   (bo =f=. 0) be polynomials prime to each other, and let C be a simple
   closed contour enclosing all zeros of Q(z). Prove that

--dz= bo

J

P(z) {

2

7riao if m - n = 1

c+ Q(z) 0 if m - n;::: 2

9. Let f be analytic, except for a finite number of isolated singularities,

in a region G that contains a deleted neighborhood of oo. Consider a

simple closed contour C in G not passing through any singularity off

and outside of which f is analytic except for the singularities bi, b 2 ,

... , bm (bk=/: oo). Prove that

j f(z)dz=27riE!t~f+211"i!~f

c- k=l.

10. Evaluate: j [sin(a/z)/(z^2 + b^2 )]dz, where a,b =f=. O, c+:

c+

0 $ t $ 271", 0 < r =/: !bl. Hint: Use the result in problem 9.

z = reit

'