1550251515-Classical_Complex_Analysis__Gonzalez_

478 Chapter^7

C +(-Ci)+···+ (-Cn)· If f is of class C^1 (R) and z ER, then

Or(z)f(z) = ~ J f(()d( - Or(z) jrf J, ded71 (7.17-6)

27rz ( - z 7r J ( -z

r fl

Proof L•et Dp = {(: 1(-zl:::; p} C R,"'(p: ( - z = peit, 0:::; t:::; 271", and
Rp = R - Dp (Fig. 7.23). By formula (7.9-5) as applied to the function
!(()/(( --z) on RP, we have, with ( = e + i'IJ,

or

But

2i Jr f a_ f(O de d'IJ = J f(O d( + J f(O d(

~ J 8( ' - z M ' - z -~ ' ' - z

2ijrf _!J_ded'IJ= J f(() d(-J f(() d( (7.17-7)

J (-z (-z (-z

Rp r+ '"fp

I

f(() d( = J f(() - f(z) d( + f(z) I --5_

(-z (-z (-z

'"(p '"(p '"(p

= j !(() - f(z) d( + 27rif(z)

(-z

'"(p

By the continuity off at z, for every E > 0 there exists 8 > 0 such that
If(() - f(z)I < E whenever IC -zl = p < 8. Hence for 0 < p < 8,

J

f(()-f(z) d( :$ .:27rp=27l"E

(-z p

'"(p

R

Fig. 7.23