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    Biholomorphic Transformation

    Alternate names

    angle-preserving transformation | biholomorphic function | biholomorphic map | biholomorphic transformation | conformal map | conformal transformation

    Basic definition

    A conformal mapping is a transformation in the complex plane that preserves local angles.

    Detailed definition

    A conformal mapping, also called a conformal map, conformal transformation, angle-preserving transformation, or biholomorphic map, is a transformation w = f(z) that preserves local angles. An analytic function is conformal at any point where it has a nonzero derivative. Conversely, any conformal mapping of a complex variable which has continuous partial derivatives is analytic. Conformal mapping is extremely important in complex analysis, as well as in many areas of physics and engineering. A mapping that preserves the magnitude of angles, but not their orientation is called an isogonal mapping.

    Related terms

    analytic function | Cauchy-Riemann equations | Cayley transform | conformal projection | harmonic function | indirectly conformal mapping | isogonal mapping | Laplace's equation | Möbius transformation | quasiconformal map | Schwarz-Christoffel mapping | similar

    Educational grade level

    college level

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