4.4.5 Hadamard Factorization Theorem
Theorem 4.4.18.
Let be the Weierstrass canonical factorization of , where is entire with finite order and . Then is a polynomial of degree .
Proof.
By logarithmic differentiation and by taking subsequent derivatives, we have
By applying Proposition 4.4.3 and Lemma 4.4.8, we have
Hence, is a polynomial of degree . Choosing so that , the assertion follows.
Corollary 4.4.4.
Let be the Weierstrass canonical factorization of , where is entire with finite order . Then is a polynomial of degree .
Proof.
Let , where and are entire, , and has finite order and for . For , such that implies
Thus, by letting . Additionally, for any , , such that
because for sufficiently large ,
Hence, . Letting implies . Let where is a constant, so that . It is also trivial that . Explicitly, we have .
By Theorem 4.4.18 on , is a polynomial of degree , and so is .
Then the results of Corollary 4.4.4 and Theorem 4.4.16 may be consolidated into a single statement:
Theorem 4.4.19 (Hadamard Factorization Theorem).
Let be the genus of and let be the order of , where is entire with finite order. Then .
Theorem 4.4.20.
The factorization
defines an entire function and uniformly converges on any compact disk .
Proof.
The zeros of are simple at each of . Aside from the simple zero at , let
enumerate the zeros of . By Example 4.4.2, and the Hadamard Factorization Theorem (Theorem 4.4.19), the order of is 1, the genus does not exceed 1, and
where is a polynomial (and where the product locally uniformly converges in ). Since the partial products , where
have a single accumulation point, the subsequence converges to the same point. Since
we have
Then from we have
Since , we have