The Geometric Theory of Holomorphic Functions

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.