[Complex Analysis] Transform a column into a line

[brolaf]

[Complex Analysis] Conformal mapping of a column into a line

Homework Statement

I'm having a problem with this problem (:rofl:) where I have to transform an area of 2 circles, the one contained within the other, into a straight line.
I've managed to transform the circles using a Möbius transformation into a column bounded by 1/4<Re w<1/2 but I have no idea how to go further than that.. I know my teacher must have talked about this at some point, but I have no memory of it. Also, the course "book" we were given is just a printed handout of some 100 pages written by the teacher himself. While it covers alot, there are some gaps in it, including this.

It seems to me fairly illogical to be able to transfer a area into a straight line like this. I mean, one could of course make the column thinner so it would "look" like a line, but to transform it into an "true" infinitely thin line is beyond my comprehension of the subject. Still, I'm pretty new to this subject and I'm still having my first glances at a lot of things.. but any help with this would be greatly appreciated!

Homework Equations

Find w that transforms:
{ z $\in$ℂ |z| < 2 and |z-1| < 1 }

into:
{ z $\in$ℂ Re z > |Im z| }

The Attempt at a Solution

w₁ = $\frac{1}{2-z}$

I made this picture to describe the problem and my progress better:

Cheers,
- brolaf

 

[mighty2000]

Dear brolaf,

Thank you for sharing your thoughts and progress on this complex analysis problem. It is indeed a challenging one, but I believe with some guidance you will be able to solve it successfully.

Firstly, it is important to understand the concept of conformal mapping. A conformal mapping is a transformation that preserves angles between curves. In other words, if two curves intersect at a certain angle in the original domain, they should also intersect at the same angle in the transformed domain.

In your problem, you are asked to transform an area bounded by two circles into a straight line. This may seem counterintuitive, but it is possible with a conformal mapping. It is important to note that the straight line in the transformed domain is not truly infinitely thin, but rather a representation of an infinite line in the original domain.

Now, let's look at your attempt at a solution. Your transformation, w1 = \frac{1}{2-z}, does indeed map the two circles into a column. However, this column is not bounded by 1/4<Re w<1/2 as you stated, but rather by -1/2<Re w<1/2. This is because the transformation maps the unit circle |z|=1 to the imaginary axis in the w-plane, and the circle |z-1|=1 to the real axis in the w-plane. So, the region between these two circles is mapped to the region between the imaginary and real axes, which is a column with a width of 1.

To transform this column into a straight line, we need to apply another transformation. One possible approach is to use the logarithmic function. The transformation w2 = \ln(w1) will map the column into a strip of width \ln(2), which is approximately 0.693. This strip is centered on the real axis, so we can shift it to the right by adding a constant, say 1, to the real part of w2. This gives us the final transformation w = \ln(w1) + 1, which maps the original region onto the desired region { z \inℂ Re z > |Im z| }.

I hope this explanation helps you to better understand the concept of conformal mapping and how to approach this problem. It is always important to refer back to your course materials and seek help from your teacher when you encounter difficulties. Keep up the good work and I wish