Linear Fractional Transformation, also known as a Möbius transformation, is a function of the form f(z) = (az + b) / (cz + d) where ad - bc ≠ 0, mapping the extended complex plane to itself. It preserves the cross-ratio of four points and is instrumental in complex analysis, geometry, and dynamic systems due to its conformal properties and ability to transform circles and lines into circles or lines.

Linear Fractional Transformation

A Möbius transformation is a complex function that maps the extended complex plane onto itself, often used in complex analysis and geometry to study conformal mappings. It is characterized by its ability to transform circles and lines into other circles or lines, preserving angles and the general shape of figures.

Mobius Transformation

Function of the form f(z) = (az + b) / (cz + d)

The extended complex plane, also known as the Riemann sphere, is a compactification of the complex plane by adding a point at infinity, allowing for a unified treatment of infinity in complex analysis. This extension facilitates a more comprehensive understanding of meromorphic functions and the behavior of complex functions at infinity.

extended complex plane

cross-ratio of four points

Complex analysis is a branch of mathematics that studies functions of complex numbers and their properties, such as differentiability and integrability, which often lead to elegant and powerful results not seen in real analysis. It plays a crucial role in various fields, including engineering, physics, and number theory, due to its ability to simplify problems and provide deep insights into the nature of mathematical structures.

complex analysis

Geometry is a branch of mathematics concerned with the properties and relationships of points, lines, surfaces, and shapes in space. It encompasses various subfields that explore dimensions, transformations, and theorems to understand and solve spatial problems.

geometry

Dynamic systems are mathematical models used to describe the time-dependent behavior of complex systems in which the state evolves according to a set of rules or equations. These systems are characterized by feedback loops, nonlinearity, and the ability to adapt or change in response to external stimuli.

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