# 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 $\mathbf C$,

$$\tag{1 } 0 \dots 0 , \alpha _ {1} , \alpha _ {2} \dots$$

$$0 < | \alpha _ {k} | \leq | \alpha _ {k+} 1 | ,\ k = 1, 2 , . . . ; \ \lim\limits _ {k \rightarrow \infty } | \alpha _ {k} | = \infty ,$$

there exists an entire function with zeros at the points $\alpha _ {k}$ of this sequence and only at these points. This function may be constructed as a canonical product:

$$\tag{2 } W( z) = z ^ \lambda \prod _ { k= } 1 ^ \infty \left ( 1- \frac{z}{\alpha _ {k} } \right ) e ^ {P _ {k} ( z) } ,$$

where $\lambda$ is the multiplicity of zero in the sequence (1), and

$$P _ {k} ( z) = \frac{z}{\alpha _ {k} } + \frac{z ^ {2} }{2 \alpha _ {k} ^ {2} } + \dots + \frac{z ^ {m _ {k} } }{2 \alpha _ {k} ^ {m _ {k} } } .$$

The multipliers

$$W \left ( \frac{z}{\alpha _ {k} } ; \ m _ {k} \right ) = \left ( 1- \frac{z}{\alpha _ {k} } \right ) e ^ {P _ {k} ( z) }$$

are called Weierstrass prime multipliers or elementary factors. The exponents $m _ {k}$ are chosen so as to ensure the convergence of the product (2); for instance, the choice $m _ {k} = k$ ensures the convergence of (2) for any sequence of the form (1).

It also follows from this theorem that any entire function $f( z)$ with zeros (1) has the form

$$f( z) = e ^ {g(} z) W( z) ,$$

where $W( z)$ is the canonical product (2) and $g( z)$ 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 $D \subset \mathbf C$: Whatever a sequence of points $\{ \alpha _ {k} \} \subset D$ without limit points in $D$, there exists a holomorphic function $f$ in $D$ with zeros at the points $\alpha _ {k}$ 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 $\alpha$ of the complex space $\mathbf C ^ {n}$, $n \geq 1$, be brought into correspondence with one of its neighbourhoods $U _ \alpha$ and with a function $f _ \alpha$ which is holomorphic in $U _ \alpha$. Moreover, suppose this is done in such a way that if the intersection $U _ \alpha \cap U _ \beta$ of the neighbourhoods of the points $\alpha , \beta \in \mathbf C ^ {n}$ is non-empty, then the fraction $f _ \alpha / f _ \beta \neq 0$ is a holomorphic function in $U _ \alpha \cap U _ \beta$. Under these conditions there exists an entire function $f$ in $\mathbf C ^ {n}$ such that the fraction $f / f _ \alpha$ is a holomorphic function at every point $\alpha \in \mathbf C ^ {n}$. This theorem is known as Cousin's second theorem (see also Cousin problems).

How to Cite This Entry:
Weierstrass theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weierstrass_theorem&oldid=49192
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article