Conformal map

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In mathematics, a conformal map is a function which preserves angles. In the most common case the function is between domains in the complex plane.

More formally, a map,

is called conformal (or angle-preserving) at \scriptstyle u_0 if it preserves oriented angles between curves through \scriptstyle u_0 with respect to their orientation (i.e., not just the acute angle). Conformal maps preserve both angles and the shapes of infinitesimally small figures, but not necessarily their size.

The conformal property may be described in terms of the Jacobian derivative matrix of a coordinate transformation. If the Jacobian matrix of the transformation is everywhere a scalar times a rotation matrix, then the transformation is conformal.

Conformal maps can be defined between domains in higher dimensional Euclidean spaces, and more generally on a Riemannian or semi-Riemannian manifold.


Complex analysis

An important family of examples of conformal maps comes from complex analysis. If U is an open subset of the complex plane, \scriptstyle\mathbb{C}, then a function

is conformal if and only if it is holomorphic and its derivative is everywhere non-zero on U. If f is antiholomorphic (that is, the conjugate to a holomorphic function), it still preserves angles, but it reverses their orientation.

The Riemann mapping theorem, one of the profound results of complex analysis, states that any non-empty open simply connected proper subset of \scriptstyle\mathbb{C} admits a bijective conformal map to the open unit disk in \scriptstyle\mathbb{C}.

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