Abstract
The main objects of study in this chapter are holomorphic functions h: U→ V, with U and V open in ℂ, that are one-to-one and onto. Such a holomorphic function is called a conformal (or biholomorphic) mapping. The fact that h is supposed to be one-to-one implies that h’ is nowhere zero on U [remember that if h’ vanishes to order k ≥ 0 at a point P ∈ U, then h is (k+1)-to-1 in a small neighborhood of P—see §§5.2.1]. As a result, h-1: V →U is also holomorphic—as we discussed in §§5.2.1. A conformal map h: U → V from one open set to another can be used to transfer holomorphic functions on U to V and vice versa: that is, f : V → ℂ is holomorphic if and only if f o h is holomorphic on U; and g : U → ℂ is holomorphic if and only if g o h-1 is holomorphic on V.
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© 1999 Springer Science+Business Media New York
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Krantz, S.G. (1999). The Geometric Theory of Holomorphic Functions. In: Handbook of Complex Variables. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-1588-2_6
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DOI: https://doi.org/10.1007/978-1-4612-1588-2_6
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-7206-9
Online ISBN: 978-1-4612-1588-2
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