Weierstrass Product Theorem

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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 $r \in \R_{> 0}$.


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

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

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



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.


Also see

Source of Name

This entry was named for Karl Theodor Wilhelm Weierstrass.