Bounds for Complex Exponential

Theorem
Let $\exp$ denote the complex exponential.

Let $z\in\C$ with $|z|\leq\frac12$.

Then $\frac12|z| \leq \left| \exp(z) -1 \right| \leq \frac32|z|$.

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

Thus

By the Triangle Inequality:
 * $\frac12 \vert z\vert\leq \left\vert \exp z - 1 \right\vert \leq \frac32\vert z\vert$

Also see

 * Bounds for Complex Logarithm