The Riemann Mapping Theorem states that any non-empty simply connected open subset of the complex plane, which is not the entire plane, can be conformally mapped onto the open unit disk. This theorem is foundational in complex analysis, illustrating the profound flexibility of holomorphic functions in transforming complex domains while preserving angles.

Riemann Mapping Theorem

non-empty simply connected open subset

The complex plane is a two-dimensional plane used to represent complex numbers, where the horizontal axis represents the real part and the vertical axis represents the imaginary part. This visualization allows for a geometric interpretation of complex number operations, such as addition, multiplication, and finding magnitudes and angles.

complex plane

conformally mapped

The open unit disk is the set of all points in the complex plane whose distance from the origin is less than one, often used in complex analysis and functional analysis. It is a fundamental example of a metric space and serves as a domain for various holomorphic functions and transformations, highlighting its importance in mathematical discourse.

open unit disk

foundational in complex analysis

Holomorphic functions are complex functions that are differentiable at every point in an open subset of the complex plane, and this differentiability implies that they are infinitely differentiable and analytic. This property leads to powerful results like the Cauchy-Riemann equations and Cauchy's integral theorem, making holomorphic functions central to complex analysis.

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