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

extended complex plane

The Riemann Sphere is a mathematical model that represents the extended complex plane, where every point on the complex plane corresponds to a point on the sphere, with the point at infinity being the north pole. This construction allows for a more comprehensive understanding of complex functions, particularly in the context of complex analysis and stereographic projection.

Riemann sphere

compactification of the complex plane

point at infinity

unified treatment of infinity

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.

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