Weierstrass factorization theorem

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In mathematics, the Weierstrass factorization theorem in complex analysis, named after Karl Weierstrass, asserts that entire functions can be represented by a product involving their zeroes. In addition, every sequence tending to infinity has an associated entire function with zeroes at precisely the points of that sequence.

A second form extended to meromorphic functions allows one to consider a given meromorphic function as a product of three factors: the function's poles, zeroes, and an associated non-zero holomorphic function.

1 Motivation
- 1.1 The elementary factors
2 The two forms of the theorem
- 2.1 Sequences define holomorphic functions
- 2.2 Holomorphic functions can be factored
3 References
4 See also

[edit] Motivation

The consequences of the fundamental theorem of algebra are twofold⁴. Firstly, any finite sequence, ${c n}$ , in the complex plane has an associated polynomial that has zeroes precisely at the points of that sequence:

$\,\prod_n (z-c_n).$

Secondly, any polynomial function in the complex plane, $p (z)$ , has a factorization

$\,p(z)=a\prod_n(z-c_n),$

where a is a non-zero constant and c_n are the zeroes of p.

The two forms of the Weierstrass factorization theorem can be thought of as extensions of the above to entire functions. The necessity of extra machinery is demonstrated when one considers whether the product

$\,\prod_n (z-c_n)$

defines an entire function if the sequence, ${c n}$ , is not finite. The answer is never, because the now-infinite product will not converge. Thus one cannot, in general, define an entire function from a sequence of prescribed zeroes or represent an entire function by its zeroes using the expressions yielded by the fundamental theorem of algebra.

A necessary condition for convergence of the infinite product in question is: each factor, $(z - c n)$ , must approach 1 as $n\to\infty$ . So it stands to reason that one should seek a function that could be 0 at a prescribed point, yet remain near 1 when not at that point and furthermore introduce no more zeroes than those prescribed. Enter the genius of Weierstrass' elementary factors. These factors serve the same purpose as the factors, $(z - c n)$ , above.

[edit] The elementary factors

These are also referred to as primary factors⁵.

For $n \in \mathbb{N}$ , define the elementary factors¹:

$E_n(z) = \begin{cases} (1 -z) & \mbox{if }n=0, \\ (1-z)\exp \left( \frac{z^1}{1}+\frac{z^2}{2}+\cdots+\frac{z^n}{n} \right) & \mbox{otherwise}. \end{cases}$

Their utility lies in the following lemma¹:

Lemma (15.8, Rudin) for z ≤ 1, |z| ≤ 1, n ∈ N_o

$\vert 1 - E_n(z) \vert \leq \vert z \vert^{n+1}.$

[edit] The two forms of the theorem

[edit] Sequences define holomorphic functions

Sometimes called the Weierstrass theorem ³

If $\lbrace z_i \rbrace_i \subset \mathbb{C}-\{0\}$ is a sequence such that:

$\vert z_i \vert \rightarrow \infty$ as $i \rightarrow \infty$
there is a sequence, $\lbrace p_i \rbrace_i \subset \mathbb{N}_o$ , such that for all r > 0, $\sum_{i} \left( \frac{r}{\vert z_i \vert}\right)^{1+p_i} < \infty.$

Then there exists an entire function that has (only) zeroes at every point of ${z i}$ ; in particular, P is such a function¹:

$P(z)=\prod_{i=1}^\infty E_{p_i}\left(\frac{z}{z_n}\right).$

The theorem generalizes to: sequences in open subsets (and hence regions) of the Riemann sphere have associated functions that are holomorphic in those subsets and have zeroes at the points of the sequence. ¹
Note also that the case given by the FTA is incorporated here. If the sequence, ${z i}$ is finite then setting $p i = 0$ suffices for convergence in condition 2, and we obtain: $\, P(z) = \prod_n (z-z_n)$ .

[edit] Holomorphic functions can be factored

Sometimes called the Weierstrass Product/Factor/Factorization theorem.[3] Sometimes called the Hadamard Factorization theorem; for example c.f. ⁵.

If f is a function holomorphic in a region, $Ω$ , with zeroes at every point of $\lbrace z_i \rbrace_i \subset \mathbb{C}-\{0\}$ then there exists an entire function g, and a sequence $\lbrace p_i \rbrace_i \subset \mathbb{R}_o^+$ such that:

$f(z)=e^{g(z)} \prod_{i=1}^\infty E_{p_i}(\frac{z}{z_i})$

- There is a unique factorization if $Π i z i$ is convergent¹.
- The theorem may be generalized to the space of meromorphic functions, in which case, the factorization is unique. Let f be a meromorphic function and $\lbrace z_i \rbrace_1^N, \lbrace p_i\rbrace_1^M$ be the zeroes and poles of the function, respectively; then: $f(z)=\frac{\prod_i(z-z_i)}{\prod_j(z-p_j)}$ .

[edit] References

Note 1: Rudin, W, Real and Complex Analysis, 3rd Ed, Mc Graw Hill, Boston, pp 301 - 304, 1987
Note 2: Weisstein, Eric W., Weierstrass Product Theorem at MathWorld.
Note 3: Weisstein, Eric W., Weierstrass's Theorem. at MathWorld.
Note 4: Knopp, K. "Weierstrass's Factor-Theorem", §1 in "Theory of Functions" Part II. New York: Dover, pp. 1-7, 1996.
Note 5: Boas, R. P., "Entire Functions", Academic Press Inc., New York, 1954, chapter 2.