Difference between revisions of "Weierstrass elementary factors"
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Revision as of 18:33, 24 May 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
Proposition: The following formula holds for $|z| \leq 1$: $$\left| 1-E_n(z) \right| \leq \left| z \right|^{n+1}.$$
Proof: █
