Entire Functions with Prescribed Zeros

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 a_l, a₂, a₃,... zu bilden, wofern nur die nothwendige Bedingung Lim_n=∞|a_n| =∞ erfüllt ist. (It is therefore always possible to construct a single-valued entire function G(x) with prescribed zeros a_l, a₂, a₃,... provided only that the necessary condition Lim_n=∞|a_n| =∞ is satisfied.)

— Weierstrass, Math. Werke 2, p. 97