Möbiusmuunnos

Möbiusmuunnos, also known as a bilinear transformation or linear fractional transformation, is a type of function encountered in complex analysis. It is a function of a complex variable $z$ that can be expressed in the form $f(z) = \frac{az+b}{cz+d}$, where $a, b, c, d$ are complex numbers and $ad-bc \neq 0$. The condition $ad-bc \neq 0$ ensures that the transformation is not degenerate.

These transformations are fundamental in the study of conformal mappings because they preserve angles and map circles and lines to circles and lines. The set of all Möbius transformations forms a group under composition. This group is often denoted as $PSL(2, \mathbb{C})$, the special linear group of 2x2 matrices with complex entries modulo its center.

Möbius transformations have applications in various fields, including geometry, complex analysis, and even theoretical physics. They are particularly important in the context of the Riemann sphere, which is the complex plane extended by a point at infinity. On the Riemann sphere, every Möbius transformation is a bijection.

The geometric interpretation of Möbius transformations is that they are compositions of simpler transformations: translations ($z \mapsto z+b$), dilations/rotations ($z \mapsto az$), and inversions ($z \mapsto 1/z$). The specific coefficients $a, b, c, d$ determine the nature of the transformation. For example, if $c=0$, the transformation is simply of the form $f(z) = (a/d)z + (b/d)$, which is a similarity transformation (a combination of scaling, rotation, and translation).