1550251515-Classical_Complex_Analysis__Gonzalez_

Singularities/Residues/ Applications

z=oo

sin3z

11. Res z=l ( Z -- 1 ) 2
    10. Resez + z -1
    z::iO
    12. Res z^4 sin ~
    z=O Z

675

z2m

13. Res z=O sin z sin Z .!_ 14. ~f (z -l)m (m > 0 an integer)

15. Res tan^2 z
    z=11:/2

ez

16. Res----
    z=oo z^2 (z^2 + 1)
    1 z^2

17. Rescos -- 18. Res ( 2 ) 2

z=3 Z - 3 z=oo Z + 4

19. Let f(z) = Bo + B_1z-^1 + B-2z-^2 + · · · valid for lzl > R. Find
    Resz=oo j2(z).


20. Find Resz=a f(z)g(z) assuming that f is analytic at a and that g(z)
    has a pole of order m at a with principal part

9.9 The Residue Theorem

This theorem, which is also due to Cauchy, has many important ap-

plications, as we shall see in later sections of this and subsequent

chapters.

Theorem 9.9 (Classical Form of the Residue Theorem). Let C be a

simple closed contour, and let f be analytic on and within C*, except for

a finite number of isolated singularities a 1 , a 2 , ••• , an none of which lies

on C*. Then we have

J f(z) dz= 27ri t F;,e:,, f(z)

c+ k=l

(9.9-1)

Proof Consider the circles ct: z - ak = rkeit, 0 :5 t :5 27r, k = 1, ... , n,

with radii rk small enough so that Ck n CJ= 0 (k-=? j) and c; n C* = 0

(Fig. 9.6). By applying Theorem 7.15 and Definition 9.11, we obtain

1 J 1 nj n 1 J

2 7ri f(z)dz = 2 7ri L f(z)dz = L 2 7ri f(z)dz

c+ k=l + k=l c+

ck k

n

=I:;;,~~ J(z)

k=l

(9.9-2)