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<y<\pi. 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<y<\pi, 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<x<1 onto the upper half plane y>0.

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