Biholomorphism
In the
.Formal definition
Formally, a biholomorphic function is a function defined on an
If there exists a biholomorphism , we say that U and V are biholomorphically equivalent or that they are biholomorphic.
Riemann mapping theorem and generalizations
If every
Alternative definitions
In the case of maps f : U → C defined on an open subset U of the complex plane C, some authors (e.g., Freitag 2009, Definition IV.4.1) define a conformal map to be an injective map with nonzero derivative i.e., f’(z)≠ 0 for every z in U. According to this definition, a map f : U → C is conformal if and only if f: U → f(U) is biholomorphic. Notice that per definition of biholomorphisms, nothing is assumed about their derivatives, so, this equivalence contains the claim that a homeomorphism that is complex differentiable must actually have nonzero derivative everywhere. Other authors (e.g., Conway 1978) define a conformal map as one with nonzero derivative, but without requiring that the map be injective. According to this weaker definition, a conformal map need not be biholomorphic, even though it is locally biholomorphic, for example, by the inverse function theorem. For example, if f: U → U is defined by f(z) = z2 with U = C–{0}, then f is conformal on U, since its derivative f’(z) = 2z ≠ 0, but it is not biholomorphic, since it is 2-1.
References
- Conway, John B. (1978). Functions of One Complex Variable. Springer-Verlag. ISBN 3-540-90328-3.
- D'Angelo, John P. (1993). Several Complex Variables and the Geometry of Real Hypersurfaces. CRC Press. ISBN 0-8493-8272-6.
- Freitag, Eberhard; Busam, Rolf (2009). Complex Analysis. Springer-Verlag. ISBN 978-3-540-93982-5.
- ISBN 0-534-13309-6.
- Krantz, Steven G. (2002). Function Theory of Several Complex Variables. American Mathematical Society. ISBN 0-8218-2724-3.
This article incorporates material from biholomorphically equivalent on