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

complex function

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

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

Conformal mappings are functions that locally preserve angles and the shapes of infinitesimally small figures, making them invaluable in complex analysis and applications like fluid dynamics and cartography. They are characterized by being holomorphic and having non-zero derivatives, ensuring that the local structure of the mapped domain remains unchanged except for scaling and rotation.

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