Weierstrass Product Theorem
Theorem
Let $\sequence {a_k}$ be a sequence of non-zero complex numbers such that:
- $\cmod {a_n} \to \infty$ as $n \to \infty$
Let $\sequence {p_n}$ be a sequence of non-negative integers for which the series:
- $\ds \sum_{n \mathop = 1}^\infty \size {\dfrac r {a_n} }^{1 + p_n}$
converges for every $r \in \R_{> 0}$.
Let:
- $\ds \map f z = \prod_{n \mathop = 1}^\infty \map {E_{p_n} } {\frac z {a_n} }$
where $E_{p_n}$ are Weierstrass elementary factors.
Then $f$ is entire and its zeroes 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 $\ds \prod_{n \mathop = 1}^\infty \map {E_{p_n} } {\frac z {a_n} }$ converges locally uniformly absolutely.
By Bounds for Weierstrass Elementary Factors and Weierstrass M-Test, this is the case.
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$\blacksquare$
Also see
Source of Name
This entry was named for Karl Theodor Wilhelm Weierstrass.