Bounds for Complex Exponential

Theorem
Let $\exp$ denote the complex exponential.

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

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

Proof
By definition of complex exponential:
 * $\exp z = \ds \sum_{n \mathop = 1}^\infty \frac {z^n} {n!}$

Thus

By the Triangle Inequality:
 * $\dfrac 1 2 \cmod z \le \cmod {\exp z - 1} \le \dfrac 3 2 \cmod z$

Also see

 * Bounds for Complex Logarithm