Consider the strip in the complex plane defined by $\{z\in\mathbb{C}:\Im(z)\in (0,\pi)\}$. Applying $\operatorname{exp}$ to this strip maps it onto the half plane $\{z\in\mathbb{C}:\Im(z)>0\}$?
What is a formal explanation why, other than just plotting a few points? Does the width of the strip matter, or does $\operatorname{exp}$ map similar strips, like $\{z\in\mathbb{C}:\Im(z)\in (0,1)\}$ to half planes also?