Bounds for Weierstrass Elementary Factors

Theorem
Let $E_p: \C \to \C$ denote the $p$th Weierstrass elementary factor:


 * $\map {E_p} z = \begin{cases} 1 - z & : p = 0 \\

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

Let $z \in \C$.

Some bound
Let $\cmod z \le \dfrac 1 2$.

Then:
 * $\cmod {\map {E_p} z - 1} \le 3 \cmod z^{p + 1}$

Another bound
Let $\cmod z \le 1$.

Then:
 * $\cmod {\map {E_p} z - 1} \le \cmod z^{p + 1}$

Proof of some bound
Let $\cmod z \le \dfrac 1 2$.

We may assume $p \ge 1$.

We have:
 * $\map {E_p} z = \map \exp {\map \log {1 - z} + \ds \sum_{k \mathop = 1}^p \frac {z^k} k}$

Then:

Because $p \ge 1$:
 * $2 \cmod z^{p + 1} \le \dfrac 1 2$

By Bounds for Complex Exponential:
 * $\cmod {\map {E_p} z - 1} \le 3 \cmod z^{p + 1}$

Also see

 * Weierstrass Factorization Theorem, what this is made for
 * Bounds for Complex Exponential
 * Bounds for Complex Logarithm