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Möbius transformation, 3 points

  • Thread starter usn7564
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  • #1
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Homework Statement


Find the Möbius transformation that maps
0 -> -1
1 -> infinity
infinity -> 1


Homework Equations


[tex]w = f(z) = \frac{az + b}{cz+d}[/tex]

Theorem:

Let f be any Möbius transformation. Then

i, f can be expressed as the composition of a finite sequence of translations, magnifications, rotations and inversions.

ii, f maps the extended complex plane one-to-one onto intself.

iii, f maps the class of circles and lines to itself

iv, f is conformal at every point except its pole

The Attempt at a Solution


My first idea was to attempt to solve it as a normal system of eq's but that quickly falls apart due to infinity being there. Been toying with the idea of using the fact that lines will map to lines or circles but don't quite know how to apply it.

And yes I know there's a formula for these exact types of questions but it's in the next sub chapter, book figures it's possible to do without knowing that. Just can't for the life of me figure it out.
 

Answers and Replies

  • #2
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Start with 0 -> -1. This gives you relation between b and d. Then simply think in limits, when f(z) -> inf, f(z) -> 1?
 
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  • #3
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Got it to work, was a bit too caught up with the fact that infinity was defined as a point (which I need to read up on more) which just messed with me. Just looking at the limits it wasn't actually bad at all.

Thanks.
 

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