Bounds for Complex Exponential

Theorem

Let $\exp$ denote the complex exponential.

Let $z \in \C$ with $\cmod z \le \dfrac 1 2$.

Then

$\dfrac 1 2 \cmod z \le \cmod {\exp z - 1} \le \dfrac 3 2 \cmod z$

$\exp z = \ds \sum_{n \mathop = 1}^\infty \frac {z^n} {n!}$

Thus

$\ds \cmod {\exp z - 1 - z}$

$=$

$\ds \cmod {\sum_{n \mathop = 2}^\infty \frac {z^n} {n!} }$

$\ds $

$\le$

$\ds \sum_{n \mathop = 2}^\infty \cmod {\frac {z^n} {n!} }$

$\ds $

$\le$

$\ds \sum_{n \mathop = 2}^\infty \frac {\cmod z^n} 2$

as $n \ge 2$

$\ds $

$=$

$\ds \frac {\cmod z^2 / 2} {1 - \cmod z}$

$\ds $

$\le$

$\ds \frac 1 2 \cmod z$

as $\cmod z \le \dfrac 1 2$

$\dfrac 1 2 \cmod z \le \cmod {\exp z - 1} \le \dfrac 3 2 \cmod z$

$\blacksquare$