A Möbius transformation is a function that maps points from the complex plane to itself using a specific formula. It is expressed as f(z) = \fracaz + bcz + d , where a, b, c, and d are complex numbers, and ad - bc ≠ 0 . This transformation can represent various geometric operations, such as rotations, translations, and dilations. Möbius transformations are significant in complex analysis and geometry because they preserve angles and the general structure of circles and lines. They can be visualized as transformations of the Riemann sphere, which helps in understanding complex functions and their properties.