What are Möbius Transformations and What are Their Applications?

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SUMMARY

Möbius transformations are defined as functions of the form f(z) = (az + b) / (cz + d), where the condition ad - bc ≠ 0 holds. These transformations represent the automorphism group of the Riemann sphere and are classified as rational bijections of the extended complex plane, preserving circles and straight lines. They are crucial in geometry and analysis, with significant applications in various mathematical fields. For further reading, Alan Beardon’s book on geometry and analysis provides comprehensive insights into this topic.

PREREQUISITES
  • Understanding of complex functions
  • Familiarity with the Riemann sphere
  • Knowledge of meromorphic functions
  • Basic concepts of group theory, specifically SU(2,C)
NEXT STEPS
  • Study the properties of rational functions in complex analysis
  • Explore the applications of Möbius transformations in geometry
  • Learn about the relationship between Möbius transformations and the Riemann sphere
  • Investigate the isomorphism between Möbius transformations with ad - bc = 1 and SU(2,C)
USEFUL FOR

Mathematicians, students of complex analysis, and anyone interested in the geometric applications of Möbius transformations will benefit from this discussion.

mprm86
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Could someone please explain me what are Möbius transformations, and what do they work for?
Where can I find more info about this?
Thanks in advance.
 
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They are the set of all functions of the form

f(z)= \frac{az+b}{cz+d}[math]<br /> <br /> subject to the condition the ad-bc=/=0<br /> <br /> They are the set of all &quot;rational bijections&quot; of the extended complex plane (ie allowing a point at infinity) that preserves the set of circles and straight lines.<br /> (ie they are the automorphism group of the riemann sphere)<br /> Rational means is of the form P(z)/Q(z) where P and Q are polynomials in z. <br /> <br /> There is plenty of information out there (google) and I believe there is a book by Alan Beardon about geometry and analysis that will explain this stuff. <br /> <br /> The important thing is they are a group of meromorphic functions on the extended plane and we can describe lots of geometry in terms of them. <br /> <br /> The set of transformations with ad-bc=1is isomorphic with SU(2,C).
 

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