Bounds for Weierstrass Elementary Factors
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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:
- $\map {E_p} z = \map \exp {\map \log {1 - z} + \ds \sum_{k \mathop = 1}^p \frac {z^k} k}$
Then:
\(\ds \cmod {\map \log {1 - z} + \sum_{k \mathop = 1}^p \frac {z^k} k}\) | \(=\) | \(\ds \cmod {-\sum_{k \mathop = p + 1}^\infty \frac{z^k} k}\) | Series Expansion of Complex Logarithm | |||||||||||
\(\ds \) | \(\le\) | \(\ds \sum_{k \mathop = p + 1}^\infty \frac {\cmod z^k} k\) | Triangle Inequality for Series | |||||||||||
\(\ds \) | \(\le\) | \(\ds \sum_{k \mathop = p + 1}^\infty \cmod z^k\) | because $k \ge 1$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\cmod z^{p + 1} } {1 - \cmod z}\) | Sum of Infinite Geometric Sequence | |||||||||||
\(\ds \) | \(\le\) | \(\ds 2 \cmod z^{p + 1}\) | 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
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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 $5.11$