An indirectly conformal mapping, sometimes called an anticonformal mapping, is a mapping that reverses all angles, whereas an isogonal mapping can reverse some angles and preserve others.For example, if is a conformal map, then is an indirectly conformal map, and is an isogonal mapping.
A metric defined bywhere denotes the positive harmonic functions on a domain. The part metric is invariant under conformal maps for any domain.
Let be a simply connected compact set in the complex plane. By the Riemann mapping theorem, there is a unique analytic function(1)for that maps the exterior of the unit disk conformally onto the exterior of and takes to . The number is called the conformal radius of and is called the conformal center of .The function carries interesting information about the set . For instance, is equal to the logarithmic capacity of and(2)where the equality holds iff is a segment of length . The Green's function associated to Laplace's equation for the exterior of with respect to is given by(3)for .
Letbe an analytic function, regular and univalent for , that maps conformally onto the region preserving the point at infinity and its direction. Then the function is uniquely determined and is called the transfinite diameter, sometimes also known as Robin's constant or the capacity of .
Let be a path in , , and and be the tangents to the curves and at and . If there is an such that(1)(2)for all (or, equivalently, if has a zero of order ), then(3)(4)so the complex argument is(5)As , and ,(6)
A conformal mapping, also called a conformal map, conformal transformation, angle-preserving transformation, or biholomorphic map, is a transformation 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 (Churchill and Brown 1990, p. 241).Several conformal transformations of regular grids are illustrated in the first figure above. In the second figure above, contours of constant are shown together with their corresponding contours after the transformation. Moon and Spencer (1988) and Krantz (1999, pp. 183-194) give tables of conformal..
The logarithmic capacity of a compact set in the complex plane is given by(1)where(2)and runs over each probability measure on . The quantity is called the Robin's constant of and the set is said to be polar if or equivalently, .The logarithmic capacity coincides with the transfinite diameter of ,(3)If is simply connected, the logarithmic capacity of is equal to the conformal radius of . Tables of logarithmic capacities have been calculated (e.g., Rumely 1989).
The linear fractional transformationthat maps the upper half-plane conformally onto the unit disk .
A transformation of the form(1)where , , , and(2)is a conformal mapping called a linear fractional transformation. The transformation can be extended to the entire extended complex plane by defining(3)(4)(Apostol 1997, p. 26). The linear fractional transformation is linear in both and , and analytic everywhere except for a simple pole at .Kleinian groups are the most general case of discrete groups of linear fractional transformations in the complex plane .Every linear fractional transformation except has one or two fixed points. The linear fractional transformation sends circles and lines to circles or lines. Linear fractional transformations preserve symmetry. The cross ratio is invariant under a linear fractional transformation. A linear fractional transformation is a composition of translations, rotations, magnifications, and inversions.To determine a particular linear fractional transformation, specify the..
If and are bounded domains, , are Jordan curves, and is a conformal mapping, then (respectively, ) extends one-to-one and continuously to (respectively, ).
An isogonal mapping is a transformation that preserves the magnitudes of local angles, but not their orientation. A few examples are illustrated above.A conformal mapping is an isogonal mapping that also preserves the orientations of local angles. If is a conformal mapping, then is isogonal but not conformal. This is due to the fact that complex conjugation is not an analytic function.
Let be a point in a simply connected region , where is the complex plane. Then there is a unique analytic function mapping one-to-one onto the disk such that and . The corollary guarantees that any two simply connected regions except (the Euclidean plane) can be mapped conformally onto each other.
Let be a transcendental meromorphic function, and let , , ..., be five simply connected domains in with disjoint closures (Ahlfors 1932). Then there exists and, for any , a simply connected domain such that is a conformal mapping of onto . If has only finitely many poles, then "five" may be replaced by "three" (Ahlfors 1933).