MHB Find Conformal Map to Remove Semidisk | Solutions & Explanations

[nacho-man]

Please refer to the attached image.

Ok, I'm in a bit of strife here.
I like to give in my own feedback and thoughts on particular questions so I can have one of you experts tell me where I am going wrong/right and help me,
however I have absolutely no idea with these two questions.

Could I please get some help/guidance as to what I'm supposed to be looking for/doing?

for instance, with i) I cannot picture what it looks like to have the "semidisk" removed. Is the unit disk simply: W = {|z|=1} ?

Are there any theorems in particular here?

Thank you very much for any help, sorry for the vague post. I am completely and utterly lost on this one! =_=

 

[Opalg]

http://mathhelpboards.com/attachments/analysis-50/1355d1379925000-find-conformal-map-q2.jpg

The idea in problems like this is always to build up a complicated conformal map as a composition of simpler ones. To do that, you must have at your fingertips a little library of known "simple" conformal maps. So I will assume that you already know about maps like $z\mapsto\frac{1+z}{1-z}$ that map the unit disk to a half-plane.

In this problem, you start with the set $W$ consisting of the upper half-plane that has had a semicircular bite taken out of it where the upper half of the unit disk has been removed. You want to map $W$ to the unit disk, and I would do this in several stages.

Stage 1. The map $f_1(z) = 1/z$ takes the region outside the unit disk to the interior of the disk, and you can check that it takes the part of the upper half-plane outside the unit disk to the lower half of the unit disk. So $f_1$ maps $W$ to the set $\{z\in\mathbb{C}:|z|<1,\ \text{im}(z)<0\}.$

Stage 2. Use the map $f_2$ from your "library" that takes the unit disk to a half-plane. You can check that this map takes the lower half of the unit disk to a quarter-plane or quadrant.

Stage 3. The map $f_3(z) = z^2$ expands the quadrant to a half-plane.

Stage 4. One of the maps $f_4$ from your library will map that half-plane to the unit disk.

Thus the composition $ f_4\circ f_3\circ f_2\circ f_1$ takes $W$ to the unit disk. You can find a detailed solution to a similar problem in http://www.math.umn.edu/~olver/pd_/cm.pdf (see Example 7.35).

For the second part of the problem, find a harmonic function on the unit disk having appropriate boundary values, and use the result from the first part of the problem to transport this function to a function on $W$.