What is the confirmal mapping that map right half plane onto d?
Answers
Answer:
Since is a Möbius transformation it sends circles onto circles. The Orientation Principle (see Chapter III, 3.21 p.53 on John Conway: Functions of one complex variable I. Second edition) states that if maps the circle onto the circle then everything on one side of must be send exclusively to only one side of (the sides depend on the orientation given).
In this case, we have that the unit circle is sent onto the imaginary axis (which is a circle on the Riemann Sphere), thus the imaginary axis is the boundary of , that is everything on the inside of must be send only to the left side or only to the right side of the imaginary axis.
To check what side (that is to check what orientation is taken), we only need to check where sends any point inside . Then since a point inside the unit circle , in this case is sent to a point on the right half plane , we get that must be the right half plane.
As a corollary we get that is the left half plane, and you an check it for example by computing.
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