Real and Complex Analysis (Rudin)

198 REAL AND COMPLEX ANALYSIS
where E(Z) -+ 0 as z -+ Zo and '1(w) -+ 0 as w -+ wo. Put w = f(z), and substitute
(2) into (3): If z =F Zo,
h(z) - h(zo) = [g'(f(zo» + '1(f(z))][f'(zo) + E(Z)]'
Z - Zo
(4)
The differentiability of f forces f to be continuous at Zo. Hence (1) follows
from (4).
10.4 Examples For n = 0, 1,2, ... , zn is holomorphic in the whole plane, and
the same is true of every polynomial in z. One easily verifies directly that liz
is holomorphic in {z: z =F OJ. Hence, taking g(w) = 1/w in the chain rule, we
see that iffl andf2 are in H(O) and 0 0 is an open subset of 0 in whichf2 has
no zero, thenfllf2 E H(Oo).
Another example of a function which is holomorphic in the whole plane
(such functions are called entire) is the exponential function defined in the
Prologue. In fact, we saw there that exp is differentiable everywhere, in the
sense of Definition 10.2, and that exp' (z) = exp (z) for every complex z.
10.5 Power Series From the theory of power series we shall assume only one fact
as known, namely, that to each power series .
00
L ciz - at (1)
n=O
there corresponds a number R E [0, 00] such that the series converges absolutely
and uniformly in D(a; r), for every r < R, and diverges ifz ¢ D(a; R). The "radius
of convergence" R is given by the root test:
..!.. = lim sup len 11/n.
R n"'oo
Let us say that a functionfdefined in 0 is representable by power series in 0
if to every disc D(a; r) c 0 there corresponds a series (1) which G:onverges to f(z)
for all z E D(a; r).
10.6 Theorem Iff is representable by power series in 0, then f E H(O) and f' is
also representable by power series in O. Infact, if
for z E D(a; r), then for these z we also have
00
f(z) = L ciz - a)n (1)
n=O
00
f'(z) = L ncn(z - a)n - 1. (2)
n=l
(2)