Real and Complex Analysis (Rudin)

ELEMENTARY PROPERTIES OF HOLOMORPHIC FUNCTIONS 211
a polynomial in (z - a) - 1, is called the principal part of f at a. It is clear in this
situation that 1 f(z) 1-4 00 as z -4 a.
In case (c),fis said to have an essential singularity at a. A statement equivalent
to (c) is that to each complex number w there corresponds a sequence {z.}
such that Z.-4 a andf(z.)-4 was n-4 00.
PROOF Suppose (c) fails. Then there exist r > 0, l> > 0, and a complex
number w such that 1 f(z) - wi> l> in D'(a; r). Let us write D for D(a; r) and
D' for D'(a; r). Define
g(z) = f(z) -
w
(z ED'). (1)
Then 9 E H(D') and 1 9 1 < Ill>. By Theorem 10.20, 9 extends to a holomorphic
function in D.
If g(a);/: 0, (1) shows thatfis bounded in D'(a; p) for some p > O. Hence
(a) holds, by Theorem 10.20.
If 9 has a zero of order m ~ 1 at a, Theorem 10.18 shows that
(z ED), (2)
where g1 E H(D) and g1(a) ;/: O. Also, g1 has no zero in D', by (1). Put h =
1/g1 in D. Then hE H(D), h has no zero in D, and
But h has an expansion of the form
f(z) - w = (z - a)-mh(z) (z ED'). (3)
00
h(z) = L biz - a)· (z ED), (4)
.=0
with bo ;/: O. Now (3) shows that (b) holds, with Ck = bm- k , k = 1, ... , m.
This completes the proof.
IIII
We shall now exploit the fact that the restriction of a power series
L c.(z - a)· to a circle with center at a is a trigonometric series.
10.22 Theorem If
and i/O < r < R, then
00
f(z) = L c.(z - a)· (z E D(a; R» (1)
.=0
(2)