Weierstrass Product Theorem

Let any finite or infinite set of points having no finite limit point be prescribed, and associate with each of its points a definite positive integer as its order. Then there exists an entire function which has zeros to the prescribed orders at precisely the prescribed points, and is otherwise different from zero. Moreover, this function can be represented as a product from which one can read off again the positions and orders of the zeros. Furthermore, if  is one such function, then

is the most general function satisfying the conditions of the problem, where  denotes an arbitrary entire function.

This theorem is also sometimes simply known as Weierstrass's theorem. A spectacular example is given by the Hadamard product.

REFERENCES:

Knopp, K. "Weierstrass's Factor-Theorem." §1 in Theory of Functions Parts I and II, Two Volumes Bound as One, Part II. New York: Dover, pp. 1-7, 1996.

Krantz, S. G. "The Weierstrass Factorization Theorem." §8.2 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 109-110, 1999.