Difference between revisions of "Weierstrass elementary factors"

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The Weierstrass elementary factors $E_n$ are defined for $n \in \{0,1,2,\ldots\}$ by
 
The Weierstrass elementary factors $E_n$ are defined for $n \in \{0,1,2,\ldots\}$ by
 
$$E_n(z)=\left\{ \begin{array}{ll}
 
$$E_n(z)=\left\{ \begin{array}{ll}
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=Properties=
 
=Properties=
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[[Weierstrass elementary factors inequality]]<br />
<strong>Proposition:</strong> The following formula holds for $|z| \leq 1$:
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[[Product of Weierstrass elementary factors is entire]]<br />
$$\left| 1-E_n(z) \right| \leq \left| z \right|^{n+1}.$$
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[[Weierstrass factorization theorem]]<br />
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<strong>Proof:</strong>  █
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=References=
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[[Category:SpecialFunction]]

Latest revision as of 19:31, 26 November 2016

The Weierstrass elementary factors $E_n$ are defined for $n \in \{0,1,2,\ldots\}$ by $$E_n(z)=\left\{ \begin{array}{ll} 1-z &; n=0 \\ (1-z)e^{z+\frac{z^2}{2}+\frac{z^3}{3}+\ldots+\frac{z^n}{n}} &; \mathrm{otherwise}. \end{array} \right.$$

Properties

Weierstrass elementary factors inequality
Product of Weierstrass elementary factors is entire
Weierstrass factorization theorem

References