Mobius Transformations

In summary, the conversation discusses a problem involving conformal mappings and Mobius Transformations. The goal is to find the Mobius Transformation that carries specific points to other points in a specific order, and then determine the image of a given domain under this transformation. The conversation also includes a potential strategy for finding the image.
  • #1
sgcbell2
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Hi, I am currently working on problem to do with conformal mappings and Mobius Transformations.

My problem is:
Find the Mobius Transformation which carries the points 1, i, 0 to the points ∞, 0, 1 (in precisely this order). Find the image of the domain { z : 0 < x < t} under this Mobius Transformation, where t > 0 is some fixed positive real number.

I have found this Mobius transformation to be (z-i)/i(z-1).

I thought that in order to find the image of the domain given above, I should input different values along the boundaries x=0 and x=t into this mobius transformations to see where they are mapped to. I have done this but I'm finding it hard to see the shape of the image.

Maybe I am doing something wrong?
 
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  • #2
Hello sg and welcome to PF.
I'm not really qualified re conformal mapping, but from what I know from complex numbers, I see the function you found transforms 1+0i to ##\infty##, 0+i to 0 and 0 + 0i to 1+0i.

I see x+0i is mapped to 1/(1-x) + ix/(1-x) which satisfies Re - I am = 1 , so I would expect a line I am = Re - 1
 
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  • #3
That makes a lot of sense! Thank you
 

1. What are Mobius Transformations?

Mobius Transformations, also known as Möbius transformations or linear fractional transformations, are mathematical functions that map one complex number to another complex number.

2. How are Mobius Transformations represented?

Mobius Transformations are represented using the formula: f(z) = (az + b)/(cz + d), where a, b, c, and d are complex numbers and z is the input number.

3. What makes Mobius Transformations unique?

Mobius Transformations are unique because they preserve circles and lines in the complex plane. This means that a circle or line in the input plane will be transformed into another circle or line in the output plane.

4. What are the applications of Mobius Transformations?

Mobius Transformations have many applications in mathematics, physics, and computer graphics. They can be used to solve complex equations, describe geometric transformations, and create 3D graphics and animations.

5. Are there any limitations to Mobius Transformations?

Yes, there are some limitations to Mobius Transformations. They cannot map the entire complex plane to itself, and they cannot map a point to infinity. In addition, some Mobius Transformations can have singularities or undefined points in their domains.

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