1550251515-Classical_Complex_Analysis__Gonzalez_

482 Chapter 7

for JzJ large enough, where A > 0 and m > 0 are real constants. Then if

b > a and Re z > b, we have

.f(z) =(PV) [-~ fb+ioo f(()d(l

27r Jb-ioo ( - Z

1. 1b+iA f(()d(

= --hm

27ri >.--+oo b-i>. ( - Z

(7.19-1)

Proof Choose A> max {J RezJ, J ImzJ} so that the given z is contained in
the rectangle

{(x,y): b~x~.A,-.A~y~.A}

and let C be the boundary of this rectangle described once in the positive
direction, L = {z: z = b -it, -A~ t ~.A}, and C 1 = C - L (Fig. 7.25).
By the classical Cauchy's formula

f(z) = ~ J f(()d(

. 27ri ( - z
c

= - _1 lb+i>. f(()d( + _1 J f(()d(

27ri b-i>. ( - z 27ri ( - z

(7.19-2)

C1

y ,b+iA.- A.+ iA.

I

I • z

I

I

I

I '

I

(^0) I ex b A. x
I
I
I
I
I L
I C1
I
b - iA. A. - iA.
Fig. 7.25