Real and Complex Analysis (Rudin)

302 REAL AND COMPLEX ANALYSIS
1 - Ep has a zero of order p + 1 at.z = 0, and if
( _ 1 - E,(z)
({)z)- Zp+l '
then (()(z) = 1:a. z·, with all a. ~ O. Hence I ({)(z) I :s; (()( 1) = 1 if I z I :s; 1, and this
gives the assertion of the lemma.
IIII
15.9 Theorem Let {z.} be a sequence of complex numbers such that z. "# 0 and
I z.l- 00 as n - 00. If {p.} is a sequence of nonnegative integers such that
00 (r)l+p.
L - < 00
.= 1 r.
for every positive r (where r. = I z.I), then the infinite product
P(z) =
.= il Ep.(-=-'
1 z.)
defines an entire function P which has a zero at each point z. and which has no
other zeros in the plane.
More precisely, if ex occurs m times in the sequence {z.}, then P has a zero
of order m at ex.
Condition (1) is always satisfied if p. = n -
l,for instance.
PROOF For every r, r. > 2r for all but finitely many n, hence rlr. < t for these
n, so (1) holds with 1 + p. = n.
Now fix r. If I z I :s; r, Lemma 15.8 shows that
(1)
(2)
if r. ~ r, which holds for all but finitely many n. It now follows from (1) that
the series
f 11 - Ep.(-=-) I
.= 1 z.
converges uniformly on compact sets in the plane, and Theorem 15.6 gives
the desired conclusion.
I I II
Note: For certain sequences {r.}, (1) holds for a constant sequence {P.}. It is
of interest to take this constant as small as possible; the resulting function (2) is
then called the canonical product corresponding to {z.}. For instance, if
1:1/r. < 00, we can take p. = 0, and the canonical product is simply