1550251515-Classical_Complex_Analysis__Gonzalez_

Integration

By Corollary 7.13 we have

or

f(() d( = 0

(-zo

f(()d( - ~ J f(()d(

( -zo - 27ri ( - zo ·

r(r)

= f(zo) J

27ri

--+-d(^1 J

( - Zo 27ri

r(r)

f(() - f(zo) d(

(-zo

481

r(r)

(7.18-2)

By the continuity if f at zo, given E > 0 there exists 8 > 0 such that

If(() - f(zo)I < e whenever IC -zol = r < 8. Hence

I

~ J f(()-f(zo) d(I::::; _e L(r) < E

27ri ( - zo 27rr

r(r)

so the last term in (7.18-2) tends to zero as r --+ 0.

Also, if we let zi = z 0 + reiti, z 2 = z 0 + reit^2 , we have

1 J d( 1 1t

2

• --= - dt= -(t2-ti)^1
  27ri ( - Zo 27!" ti 27!"
  r(r)

Thus (7.18-2) yields

1 j f(() d( = f(zo) (t2 -ti)+ 11(r)

27ri ( - Zo 27r

C1(r)

where 17(r) --+ 0 as r --+ O, and it follows that

(PV)-

2

1

. J ~(() d( = lim [ f(
2

zo) (t2 -ti)+ 17(r)] =

2

1

>.(zo)f(zo)

7ri - Zo r->O 7r 7r

c

7.19 CAUCHY'S INTEGRAL FORMULA FOR

FUNCTIONS ANALYTIC IN A HALF-PLANE

Theorem 7.27 Let f be analytic for Rez >a, and suppose that

A

lf(z)I < izlm