Index

Fix real constants $a,b$ with $a<b$ and consider the sets \begin{align*} A &= \{ z \in\mathbb{C} \;:\; e^a < |z| < e^b\} = \text{ annulus } \\ B &= \{ z \in\mathbb{C} \;:\; a < \text{Re}(z) < b\} = \text{ vertical band }. \end{align*} This picture illustrates the behaviour of the map $\exp\colon B\to A$. The imaginary axis runs from left to right, and the space between the red bars is $2\pi i$.