1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

300 CHAPTER 8 • RESIDUE THEORY

2. Let f and g have an isolated singularity at z 0. Show that Res[/+ g, zo) =
   Res[f , z o] + Res(g, zo].


3. Evaluate

dz

(a) J - 4- ·

Cj*"(-l+i) z + 4

(b) I 2 dz.

C i°(i) z(z - 2z+2)

(c) f e;z dz.

ct(o) z +z

(d) J si~z dz

2

_

ct(o) 4z - 7r

(e) J si~z dz.

Ci(O) z + 1

(f) I --/=--.

C {(O) z sm z

d z

(g) I -. 2 -·

C{ (O) Z Sm z

4. Let f and g be analytic at zo. If f (zo) ::J 0 and g has a simple zero at z 0 ,

then show that Res[~, zo) =: ~::~.

5. Find J (z - 1)-^2 {z^2 + 4) -

1

dz when

c

(a) c = ct (1).

(b) C =Ct (O).

6. Find J (z^6 +1) -
   1
   dz when
   c

(a) c =er (i).

(b) C =Ct (1; i). Hint: If zo is a singularity of f (z) = z 6 ~

1

, show

that Resjf,zo) = -~zo.