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 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 . This is easily accomplished by . The factor is responsible for the rotation (90 degree anticlockwise), while the factor is responsible for the scaling.
Let us consider the exponential mapping . Consider line . This line will be mapped onto the ray (from the origin) with argument . Since , the image is a collection of rays of arguments that “sweeps” across and covers the entire upper half plane.
In conclusion, the map will do the job for mapping the vertical strip onto the upper half plane .