57.1. ANALYTIC FUNCTION WITH PRESCRIBED ZEROS 1791

Since |zn| → ∞, there exists N such that for n > N, |zn| > 2R. Therefore, for |z| < R andletting 0 < a = min{|zn| : n≤ N} ,

∞

∑n=1

∣∣∣∣Epn

(zzn

)−1∣∣∣∣ ≤ 3

N

∑n=1

∣∣∣∣Ra∣∣∣∣pn+1

+3∞

∑n=N

(R2R

)pn+1

< ∞.

By the Weierstrass M test, the series in 57.1.6 converges uniformly for |z| < R and so thesame is true of the infinite product. It follows from Lemma 51.3.13 on Page 1624 that P(z)is analytic on |z|< R because it is a uniform limit of analytic functions.

Also by Theorem 57.0.2 the zeros of the analytic P(z) are exactly the points, {zn} ,listed according to multiplicity. That is, if zn is a zero of order m, then if it is listed m timesin the formula for P(z) , then it is a zero of order m for P. This proves the theorem.

The following corollary is an easy consequence and includes the case where there is azero at 0.

Corollary 57.1.5 Let {zn} be a sequence of nonzero complex numbers which have no limitpoint and let {pn} be a sequence of nonnegative integers such that

∞

∑n=1

(r|zn|

)1+pn

< ∞ (57.1.7)

for all r ∈ R. Then

P(z)≡ zm∞

∏n=1

Epn

(zzn

)is analytic Ω and has a zero at each point, zn and at no others along with a zero of order mat 0. If w occurs m times in {zn} , then P has a zero of order m at w.

The above theory can be generalized to include the case of an arbitrary open set. First,here is a lemma.

Lemma 57.1.6 Let Ω be an open set. Also let {zn} be a sequence of points in Ω which isbounded and which has no point repeated more than finitely many times such that {zn} hasno limit point in Ω. Then there exist {wn} ⊆ ∂Ω such that limn→∞ |zn−wn|= 0.

Proof: Since ∂Ω is closed, there exists wn ∈ ∂Ω such that dist(zn,∂Ω) = |zn−wn| .Now if there is a subsequence,

{znk

}such that

∣∣znk −wnk

∣∣ ≥ ε for all k, then{

znk

}must

possess a limit point because it is a bounded infinite set of points. However, this limitpoint can only be in Ω because

{znk

}is bounded away from ∂Ω. This is a contradiction.

Therefore, limn→∞ |zn−wn|= 0. This proves the lemma.

Corollary 57.1.7 Let {zn} be a sequence of complex numbers contained in Ω, an opensubset of C which has no limit point in Ω. Suppose each zn is repeated no more than finitelymany times. Then there exists a function f which is analytic on Ω whose zeros are exactly{zn} . If w ∈ {zn} and w is listed m times, then w is a zero of order m of f .