Abstract
If f ≠ 0 is a holomorphic function on a domain G,its zero set Z(f) is locally finite in G by the identity theorem (cf. I.8.1.3). It is natural to pose the following problem:
Let T be any locally finite subset of G, and let every point d ∈ T be assigned a natural number ∂(d) ≥ 1 in some way. Construct functions holomorphic in G which each have zero set T and, moreover, whose zeros at each point d E T have order ∂(d).
Es ist also stets möglich, eine ganze eindeutige Function G(x) mit vorgeschriebenen Null-Stellen al, a2, a3,... zu bilden, wofern nur die nothwendige Bedingung Limn=∞|an| =∞ erfüllt ist. (It is therefore always possible to construct a single-valued entire function G(x) with prescribed zeros al, a2, a3,... provided only that the necessary condition Limn=∞|an| =∞ is satisfied.)
— Weierstrass, Math. Werke 2, p. 97
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Remmert, R. (1998). Entire Functions with Prescribed Zeros. In: Classical Topics in Complex Function Theory. Graduate Texts in Mathematics, vol 172. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-2956-6_3
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