Weierstrass factor theorem | Weierstrass factorization 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 G_0(z) is one such function, then G(z) = e^(h(z)) G_0(z) is the most general function satisfying the conditions of the problem, where h(z) denotes an arbitrary entire function.

Hadamard product | Weierstrass's theorem