ABSTRACT
In this book, we will study functions defined inside and on the boundary of a bounded domain Ω in the plane with C∞ smooth boundary. Such a domain is finitely connected and its boundary consists of finitely many C∞ smooth simple closed curves. We let bΩ denote the boundary of Ω. If Ω is n-connected, there are C∞ complex valued functions zj(t) of t ∈ [0, 1] ⊂ R, j = 1, . . . n, that parameterize the n boundary curves of Ω in the standard sense. This means that, if zj(t) parameterizes the j-th boundary curve of Ω, it is understood that zj(t) and all its derivatives agree at the endpoints t = 0 and t = 1, that z′j(t) is nowhere vanishing, and that zj(t) traces out the curve exactly once. Furthermore, −iz′j(t) is a complex number representing the direction of the outward pointing normal vector to the boundary at the point zj(t). We say that a function g defined on the boundary of Ω is C∞ smooth on bΩ if, for each j, g(zj(t)) is a C
∞ function of t on [0, 1], all of whose derivatives agree at the endpoints 0 and 1. This definition seems to depend on the choice of the parameterization functions zj(t), but it is an easy exercise to see that it in fact does not. We let C∞(bΩ) denote the space of C∞
functions on bΩ. One of the most important C∞ smooth functions on the boundary of Ω is the complex unit tangent function T . If z ∈ bΩ, then T (z) is equal to the complex number of unit modulus that represents the direction of the tangent vector to bΩ at z pointing in the direction of the standard orientation of the boundary. To be precise, T is characterized by the formula T (zj(t)) = z
′ j(t)/|z′j(t)|. Since the differential dz is given
by dz = z′j(t) dt and the differential ds of arc length is given by ds = |z′j(t)| dt, we see that dz = T ds.
