Complex Variable: Conformal Mappings

[ReyChiquito]

Hello, i have this q.

The problem is that i need to transform the region 0<Re(z)<pi/2 into the unit circle. Now here is what I've done.

First i transform z_1=iz (rotate pi/2)
then z_2=Im(z_1)/(pi/2-Im(z_1)) (expand the band onto the whole upper plane)
and then z_3=(1/2+iz_2)/(1/2-iz_2) (transform the upper plain into the unit circle).

Now, substituting i get w(z)=(pi/2-Re(z)(1-2i))/(pi/2-Re(z)(1+2i)).

Is this correct?

Ive shown that the transformation is 1-1. Is this enough to state that is conformal or do i need to prove something else (that it is analytical, which is not clear for me). If so, what criteria would you suggest (less work)?

Plus, i need to do the same with the intersection of two discs (r=1 centers (1,0) and (0,1)) and in that case i have no clue how... should i send one part of the boundary to y=0 and the other to y=infinitum and then do the last transform or there is an easier way?

ps. sorry for bad english

 

[ReyChiquito]

pss... never mind... is much easier with the exponential function

i can't believe nobody tried to help me though... isn't a lot of mathematicians here?

did i post this in the wrong place?

 

[M98Ranger]

Hello there,

The steps you have taken to transform the region 0<Re(z)<pi/2 into the unit circle seem correct. You have successfully shown that the transformation is one-to-one, which is one of the criteria for a conformal mapping. However, to prove that it is conformal, you also need to show that it preserves angles. This can be done by considering a small angle in the original region and showing that it is equal to the angle in the transformed unit circle.

To prove that the mapping is analytical, you need to show that it is differentiable at every point in the region. This can be done by taking the derivative of the mapping function and showing that it exists and is continuous at every point in the region. This may require some additional work, but it is necessary to fully prove that the mapping is conformal.

As for the intersection of two discs, there are multiple ways to approach this problem. One possible approach is to use the same steps you used for the first problem, but with a different mapping function. Another approach would be to use a conformal mapping that is specific to this problem, such as the Schwarz-Christoffel mapping. This may require more work, but it will provide a more direct solution. It is up to you to decide which approach is more suitable for your problem.

I hope this helps! Good luck with your problem.