1550251515-Classical_Complex_Analysis__Gonzalez_

490 Chapter 7

Proof We have

F(n)( z ) = n. I J (( f(() - z)n+l d(

'Y

for z E Ro and n = 1, 2,.... This formula applies also to F(z) if we

define p(o)(z) = F(z) and O! = 1. Since f is continuous along"( we have

J f ( () J S M for all ( E "f. Suppose that "f is contained in the disk J ( J < K.

Then, by Darboux's inequality, we obtain, for JzJ > K.

I

(n)( )J I ML("f)

F z S n. (JzJ - K)n+i

Since the right-hand side tends to zero as z --+ oo in Ro, the conclusion
follows.

7.21 Cauchy's Formulas for the Derivatives

If f is analytic on and within the simple closed contour C we have,

by (7.17-1),

D.c(z)f(z) = ~ J J(()d(

27ri ( - z

c

whenever z r:/. C*. Hence as particular cases of (7.20-7), we have

n ( )!'( ) = ~ j J ( O d(

c z z 27ri ( ( - z )2

c

" ( )!"( ) = ~ J J(()d(

HG z z 2?ri ( ( - z )3

c

D.c(z)f(n)(z) = ~ J J(() d(

27ri ((-z)n+l

(7.21-1)

c

If z E Int C* and D.c(z) = +1, the formulas above become

J'(z) = ~ J J(() d(

27ri ((-z)2

c+

J"(z) = ~ J J(() d(

27ri ((-z)3

c+