Linear Fractional Transformation

[Hertz]

Homework Statement

I'm given two circles in the complex plane. $|z|=1$ and $|z-1|=\frac{5}{2}$. The goal is to find a "Linear Fractional Transformation" or Mobius Transformation that makes these two circles concentric about the origin.

Homework Equations

$w=f(z)=\frac{az+b}{cz+d}$

The Attempt at a Solution

From other examples I've seen of this, people typically pick three points on the curve they are trying to transform and 3 points on the curve they are trying to transform too. They then use the equation above and solve for the coefficients.

The problem for me is that I don't know exactly what I'm mapping my plane too. I don't know what w values correspond with what z values. All I know is that I am trying to make the two circles concentric about the origin.

Please please please don't beat around the bush. I have a final in 2.5 hrs that may have this material on it. We didn't cover it in class so I think it's ridiculous that it may be on the final, but it was on the practice final so I need to learn it now. You don't have to just give me the answer, but please at least just tell me how to get it. I can take it from there

 

[scurty]

Homework Statement

I'm given two circles in the complex plane. $|z|=1$ and $|z-1|=\frac{5}{2}$. The goal is to find a "Linear Fractional Transformation" or Mobius Transformation that makes these two circles concentric about the origin.

Homework Equations

$w=f(z)=\frac{az+b}{cz+d}$

The Attempt at a Solution

From other examples I've seen of this, people typically pick three points on the curve they are trying to transform and 3 points on the curve they are trying to transform too. They then use the equation above and solve for the coefficients.

The problem for me is that I don't know exactly what I'm mapping my plane too. I don't know what w values correspond with what z values. All I know is that I am trying to make the two circles concentric about the origin.

Please please please don't beat around the bush. I have a final in 2.5 hrs that may have this material on it. We didn't cover it in class so I think it's ridiculous that it may be on the final, but it was on the practice final so I need to learn it now. You don't have to just give me the answer, but please at least just tell me how to get it. I can take it from there

Here's a quick run down of what is happening in your case. If you know three distinct points map to three other distinct points under your LFT, your LFT is unique. Also, when dealing with circles, a way to construct your LFT is to map a point on one circle to a point on the other circle, map the center of the first circle to the center of the second circle, and then map infinity to infinity (since they are both circles).

so, in your case:

$T(0) \mapsto 1$ (center to center)
$T(1) \mapsto \frac{7}{2}$ (point on circle to point on circle)
$T(\infty) \mapsto \infty$ (outside point to outside point)

Start with the LFT definition and plug in 0,1,infinity and find the values of a,b,c,d