Weierstrass Product Theorem

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Theorem

Let $\left\langle{a_k}\right\rangle$ be a sequence of non-zero complex numbers such that:

$\left\vert{a_n}\right\vert \to \infty$ as $n \to \infty$

Let $\left\langle{p_n}\right\rangle$ be a sequence of non-negative integers for which the series:

$\displaystyle \sum_{n \mathop = 1}^\infty \left\vert{\dfrac r {a_n} }\right\vert^{1 + p_n}$

converges for every $r \in \R_{> 0}$.

Let:

$\displaystyle f \left({z}\right) = \prod_{n \mathop = 1}^\infty E_{p_n} \left({\frac z {a_n} }\right)$

where $E_{p_n}$ are Weierstrass elementary factors.


Then $f$ is entire and its zeros are the points $a_n$, counted with multiplicity.


Proof

By:

Locally Uniformly Absolutely Convergent Product is Locally Uniformly Convergent
Infinite Product of Analytic Functions is Analytic
Zeroes of Infinite Product of Analytic Functions

it suffices to show that the product $\displaystyle \prod_{n \mathop = 1}^\infty E_{p_n} \left({\frac z {a_n} }\right)$ converges locally uniformly absolutely.

By Bounds for Weierstrass Elementary Factors and Weierstrass M-Test, this is the case.


$\blacksquare$


Also see


Source of Name

This entry was named for Karl Theodor Wilhelm Weierstrass.