What are Möbius Transformations and What are Their Applications?

[mprm86]

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.

 

[matt grime]

They are the set of all functions of the form

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

subject to the condition the ad-bc=/=0

They are the set of all "rational bijections" of the extended complex plane (ie allowing a point at infinity) that preserves the set of circles and straight lines.
(ie they are the automorphism group of the riemann sphere)
Rational means is of the form P(z)/Q(z) where P and Q are polynomials in z.

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.

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.

The set of transformations with ad-bc=1is isomorphic with SU(2,C).\)

 

[tacman]

Möbius transformations, also known as linear fractional transformations, are a type of mathematical transformation that maps the complex plane onto itself. They are named after the German mathematician August Ferdinand Möbius who first studied them in the 19th century.

In simple terms, Möbius transformations involve dividing two complex numbers and then adding a constant. They can be expressed as the formula f(z) = (az + b) / (cz + d), where a, b, c, and d are complex numbers and z is a complex variable.

Möbius transformations have many applications in mathematics, physics, and engineering. They are particularly useful in complex analysis, as they preserve the shape of geometric figures and allow for easy calculation of integrals and derivatives.

You can find more information about Möbius transformations in various mathematical textbooks and online resources. Some recommended sources include "Complex Variables and Applications" by James Ward Brown and Ruel V. Churchill, and "Möbius Transformations in Geometry" by Oleg A. Belyaev. Additionally, there are many online tutorials and videos available that explain Möbius transformations in detail.