the prescribed-zeros theorem

Suppose you write down any wish-list of zeros: a sequence of points a_1, a_2, ... in the plane, possibly with repetitions to indicate multiplicity, subject only to the demand that they do not pile up — they march off to infinity, |a_n| -> infinity. Question: is there an entire function whose zeros are EXACTLY those points, with exactly those multiplicities, and no others? The prescribed-zeros theorem answers yes, always. This is the existence half of Weierstrass's theory, stated as a freedom: you may dictate the zeros of an entire function as freely as you like, provided they have no finite accumulation point.

The construction is exactly the Weierstrass product. For each a_n you take an elementary factor E_(p_n)(z/a_n), which contributes a single zero at a_n and nothing else, and you choose the degrees p_n growing just fast enough that the product converges uniformly on compacta. (A degree p_n = n always works, regardless of how the a_n behave.) Throw in a factor z^m if you also want a zero of order m at the origin. The resulting product is entire and vanishes precisely on your list. The one structural constraint — that the zeros cannot accumulate at a finite point — is forced on you, not by Weierstrass, but by the identity theorem: an entire function whose zeros pile up somewhere finite would have to be identically zero.

This is a striking statement of how flexible holomorphic functions are. On the real line you cannot in general find a real-analytic function with an arbitrary infinite zero set; in the complex plane you always can, as long as the set is discrete. It also frees the later constructions: once you can build a function g with any prescribed zeros and any prescribed poles (using Mittag-Leffler for the poles), you can build a meromorphic function with any prescribed zeros and poles by taking quotients — the full design freedom of function theory rests on this theorem.