1550251515-Classical_Complex_Analysis__Gonzalez_

Singularities/Residues/Applications 733

3. c;t; is the positively oriented boundary of the rectangle with vertices

at (b + m, -m), (b + m, m), (-b - m, m), and (-b - m, -m), where

b ;::: 0, b =/:-Re a, and m is a positive integer (Fig 9.31 ).

4. lf(z)I < K (a constant) for z E c;t; as m --+ 00

5. Sm = sum of the residues or f(z)g(z) at the singularities or g(z)
   enclosed by c;t;.

Then

S = - L ,Bng( a + n)

where S = limm-+oo Sm, and the summation on the right contains only

terms corresponding to those values of n for which f has a pole and g is

analytic.

Proof The residue theorem gives
I
~ j f(z)g(z) dz=" ,Bng(a + n) +Sm
27ri L.J
(9.12-1)
c;!;,

the first sum on the right extending over those poles of f enclosed by c;t;

where g is analytic. In view of assumption 2, given any E > 0, there is

M > 0 such that lzl > M implies that lzg(z)I < E. Also, for z E C;t; we

have lzl ;::: m and L(C;t;) =Sm+ 2b. Hence

j f(z)g(z) dz

c+ m

y

....

i

a-3 a-2 a-1

-b-m^0

Fig. 9.31

= j f(z)zg(z) ~z :::; :f (Sm+ 4b)

c;!;,

b

= KE(S +4-) < 12KE

m

m -

.....
a a+1 a+2 a+3

b+m x

r

m c+ m