# Bounds for Weierstrass Elementary Factors

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## Contents

## 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

### Proof of some bound

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

We may assume $p \ge 1$.

We have:

- $\map {E_p} z = \map \exp {\map \log {1 - z} + \displaystyle \sum_{k \mathop = 1}^p \frac {z^k} k}$

Then:

\(\displaystyle \cmod {\map \log {1 - z} + \sum_{k \mathop = 1}^p \frac {z^k} k}\) | \(=\) | \(\displaystyle \cmod {-\sum_{k \mathop = p + 1}^\infty \frac{z^k} k}\) | Series Expansion of Complex Logarithm | ||||||||||

\(\displaystyle \) | \(\le\) | \(\displaystyle \sum_{k \mathop = p + 1}^\infty \frac {\cmod z^k} k\) | Triangle Inequality for Series | ||||||||||

\(\displaystyle \) | \(\le\) | \(\displaystyle \sum_{k \mathop = p + 1}^\infty \cmod z^k\) | because $k \ge 1$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac {\cmod z^{p + 1} } {1 - \cmod z}\) | Sum of Geometric Sequence | ||||||||||

\(\displaystyle \) | \(\le\) | \(\displaystyle 2 \cmod z^{p + 1}\) | because $\cmod z \le \dfrac 1 2$ |

Because $p \ge 1$:

- $2 \cmod z^{p + 1} \le \dfrac 1 2$

By Bounds for Complex Exponential:

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

$\blacksquare$

### Proof of another bound

## Also see

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

## Sources

- 1973: John B. Conway:
*Functions of One Complex Variable*$VII$: Compact and Convergence in the Space of Analytic Functions: $\S5$: Weierstrass Factorization Theorem: Lemma $5.11$