Definition:Weierstrass Elementary Factor

Definition

Let $n \in \Z_{\ge 0}$ be a positive integer.

The $n$th (Weierstrass) elementary factor is the function $E_n: \C \to \C$ defined as:

$\map {E_n} z = \begin {cases} 1 - z & : n = 0 \\

\paren {1 - z} \map \exp {z + \dfrac {z^2} 2 + \cdots + \dfrac{z^n} n} & : \text{otherwise} \end {cases}$

Also see

• Bounds for Weierstrass Elementary Factors, which motivates their definition
• Weierstrass Factorization Theorem

Source of Name

This entry was named for Karl Theodor Wilhelm Weierstrass.

Sources

• 1973: John B. Conway: Functions of One Complex Variable ... (previous) $VII$: Compact and Convergence in the Space of Analytic Functions: $\S5$: Weierstrass Factorization Theorem: Definition $5.10$