# Weierstrass theorem

There are a number of theorems named after Karl Theodor Wilhelm Weierstrass (1815-1897).

### Infinite product theorem

Weierstrass' infinite product theorem : For any given sequence of points in the complex plane , (1) there exists an entire function with zeros at the points of this sequence and only at these points. This function may be constructed as a canonical product: (2)

where is the multiplicity of zero in the sequence (1), and The multipliers are called Weierstrass prime multipliers or elementary factors. The exponents are chosen so as to ensure the convergence of the product (2); for instance, the choice ensures the convergence of (2) for any sequence of the form (1).

It also follows from this theorem that any entire function with zeros (1) has the form where is the canonical product (2) and is an entire function (see also Hadamard theorem on entire functions).

Weierstrass' infinite product theorem can be generalized to the case of an arbitrary domain : Whatever a sequence of points without limit points in , there exists a holomorphic function in with zeros at the points and only at these points.

The part of the theorem concerning the existence of an entire function with arbitrarily specified zeros may be generalized to functions of several complex variables as follows: Let each point of the complex space , , be brought into correspondence with one of its neighbourhoods and with a function which is holomorphic in . Moreover, suppose this is done in such a way that if the intersection of the neighbourhoods of the points is non-empty, then the fraction is a holomorphic function in . Under these conditions there exists an entire function in such that the fraction is a holomorphic function at every point . This theorem is known as Cousin's second theorem (see also Cousin problems).