Möbiusmuunnosten

Möbiusmuunnosten are a type of conformal transformation of the complex plane. They are also known as linear fractional transformations or fractional linear transformations. A Möbius transformation is a function of the form f(z) = (az + b) / (cz + d), where a, b, c, and d are complex numbers and ad - bc is not equal to zero. The condition ad - bc != 0 ensures that the transformation is indeed a transformation and not a constant.

These transformations have several important properties. They map generalized circles (circles and lines) in the complex plane to generalized circles. They preserve angles, which is why they are called conformal mappings. The set of all Möbius transformations forms a group under composition, known as the Möbius group. This group is fundamental in complex analysis and has connections to hyperbolic geometry and the study of Riemann surfaces.

Möbius transformations can be understood geometrically as acting on the Riemann sphere, which is the complex plane augmented with a point at infinity. On the Riemann sphere, every Möbius transformation is an inversion followed by reflections or translations. The study of Möbius transformations is crucial for understanding the geometry and topology of the complex plane and related spaces. They appear in various areas of mathematics, including complex analysis, geometry, and number theory.