568 Chapters
Hence
lf(z) - f(zo)I:::; lz - zolThus for any given f > 0 we have lf(z)-f(zo)I < f provided that lz-zol <
s = f,
II. All the preceding properties remain valid for functions of class 'D
as well as for functions of class 1i(D), since both classes are contained
in the class of continuous functions. In the next section we discuss those
properties pertaining more specifically to the class of analytic functions.
8.9 ZJEROS OF ANALYTIC FUNCTIONS. IDENTITY
PRINCIPLE FOR ANALYTIC FUNCTIONS
Theorem 8.20 A zero of finite order m ~ 2 of an analytic function J:
D -+ C, D open, is a zero of order m - 1 of its derivative f'. A simple
zero of f is not a zero of f'.
Proof From (8.8-1)
J(z) = (z - zorg(z)
where g(z) is now analytic in D and g(zo) f 0, we obtain
J'(z) = (z - zor-^1 [mg(z) + (z - zo)g'(z)]
= (z - zor-^1 h(z) (8.9-1)where h(z) = mg(z)+(z-zo)g'(z) is analytic in D, and h(zo) = mg(zo) f 0.
Hence it follows that zo is a zero of order m - 1 off' if m ~ 2. However,
if m = 1, then f'(z) = h(z) and J'(zo) = h(zo) = g(zo) f 0, so that zo
is not a zero of f'.
Corollary 8.10 If zo is a zero of order m ~ 1 of an analytic function,
f: D -+ C, D open, then f(zo) = f'(zo) = · · · = j(m-l)(zo) = 0 while
j(m)(z 0 ) f 0, and conversely.
In the statement above we make use of the convention j(^0 )(z 0 ) = f(z 0 ).
Proof The first part follows at once by repeated application of Theo-
rem 8.20. To prove the converse, by the Cauchy-Taylor expansion of f(z)
about zo we have
f'(zo) J(m-l)(zo) m-1
f(z) = f(zo) + ----rr-(z - zo) + · · · + (m _ l)! (z - zo)f(m)(zo)
+ m! (z -zor +... (8.9-2)