Mapping of Infinite Vertical Strip by the Exponential Function

This post is continued from a previous post on how to map an open region between two circles to a vertical strip. We wish to map the infinite vertical strip $0<\text{Re}(z)<1$ onto the upper half plane. Reference book is Complex Variables and Applications, Example 3 page 44.

First, we need to map the vertical strip onto the horizontal strip $0. This is easily accomplished by $w=\pi iz$. The factor $i$ is responsible for the rotation (90 degree anticlockwise), while the factor $\pi$ is responsible for the scaling.

Let us consider the exponential mapping $w=e^z=e^{x+iy}=e^x\cdot e^{iy}$. Consider line $y=k$. This line will be mapped onto the ray (from the origin) with argument $k$. Since $0, the image is a collection of rays of arguments $0<\theta<\pi$ that “sweeps” across and covers the entire upper half plane.

In conclusion, the map $f(z)=e^{\pi i z}$ will do the job for mapping the vertical strip $0 onto the upper half plane $y>0$.