1550251515-Classical_Complex_Analysis__Gonzalez_

496 Chapter 7

Proof By (7.21-2) we have

j<n)(zo)= ~ J f(()d(

21l"i ((-zo)n+l

c+

This formula holds also for n = 0 with the notational conventions men-

tioned in the proof of Theorem 7.29. The same conventions will be used

elsewhere in this book. On applying Darboux's inequality we get

IJ(n)(zo)I:::; n! ~ (211"r) = Mn!

211" 1 n+l 1 n

The example f(z) = zn with z 0 = 0 and r = 1 shows that inequal-

ity (7.2:l-1) cannot be improved since in this case M = 1 and J(n)(O) =

n!.

7.24 CAUCHY-LIOUVCLLE THEOREM

Theorem 7.33 If f is analytic in C (i.e., if f is an entire function), and

if IJ(z)I :::; M for all z E C, then f is a constant function in C.

Proof With the assumptions of this theorem we may apply Cauchy's
inequality (7.23-1) for n = 1 to any point z 0 E C, so that

IJ'(zo)I:::; M

r

(7.24-1)

whatever be the radius r of the circle. Since the right-hand side of (7.24-1)

can be made as small as we please by taking r large enough, we see that

f' ( z 0 ) = 0. This being true for an arbitrary point z 0 , we have f' ( z) = 0 for

all z EC. But this implies that J(z) = k (a constant) in C by Theorem 6.6.

Remarks This important theorem furnishes brief proofs of a number of

theorems in complex analysis. As an example, in the next section we prove

the so-called fundamental theorem of algebra.

Theo1rem 7.33 is also due to Cauchy (8], but was attributed to Liouville

by Borchardt, who learned it from Liouville in his 184 7 lectures (according

to E. T. Whittaker and G. N. Watson (39], p. 105).

Corolla1ry 7.16 If f is a nonconstant analytic function in C, then for any

M > 0 we have

lf(z)I > M (7.24-2)

for some values of z E C.

Example Let f(z) = ez, z = x + iy. We have IJ(z)I =ex> M whenever

x = Rez > lnM.