ABSTRACT
The main objects of study in this chapter are holomorphic functions h : U → V, with U and V open domains 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 ≥ 1 at a point P ∈ U , then h is (k + 1)-to-1 in a small neighborhood of P-see Section 6.2.1]. As a result, h−1 : V → U is also holomorphic-as we discussed in Section 6.2.1. 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 ◦ h is holomorphic on U ; and g : U → C is holomorphic if and only if g ◦ h−1 is holomorphic on V.
