Möbius Transformations

Möbius transformations are special functions that map the complex plane onto itself using the formula \(f(z) = \frac{az + b}{cz + d}\), where \(a, b, c, d\) are complex numbers. These transformations preserve angles and the general shape of figures, such as circles and lines, although they can rotate, stretch, or invert them. They are a key concept in complex analysis, helping to understand how geometric structures behave under such mappings and playing a role in fields like geometry, physics, and engineering.