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.
