1550251515-Classical_Complex_Analysis__Gonzalez_

Functions. Limits and Continuity. Arcs and Curves 145 lim Argf(z) = Arg(lim f(z)) = Argf(a) z~a z~a provided that J(a) + \J(a)\ ...

146 Chapter 3 Corollary 3.1 Any polynomial function P(z) = a 0 zn + a1zn-l +···+an is a continuous function in (<C,d). A rati ...

Functions. Limits and Continuity. Arcs and Curves 147 Proof Let A be any open set in W. We have By the continuity of g, g-^1 (A) ...

148 Chapter^3 are open, and Q* = {f-^1 (A): A E Q} is a covering of E. Since E is compact, there is a finite number of sets in Q ...

Functions. Limits and Continuity. Arcs and Curves 149 Hence d'(f(x),l(x')):::; d'(f(x),l(x;)) + d'(f(x;),l(x')) < l/2€ + l/2€ ...

150 Chapter^3 For example, an arbitrary open interval of the real line JR is homeomorphic to JR. A mapping f: X ---7 Y is a loca ...

Functions. Limits and Continuity. Arcs and Curves 151 However, in what follows we restrict our attention to oriented arcs whose ...

152 Chapter^3 positive or natural orientation of the interval [a, ,B] (as defined by the "less than" relation) induces on the gr ...

Functions. Limits and Continuity. Arcs and Curves 153 y (1, 1) x 0 Y2 Fig. 3.14 the initial point of the graph under (a) or (b) ...

154 Chapter^3 where a = h(a'), (J = h((J'), have the same oriented graph. Because of this, they share some common properties. Fo ...

Functions. Limits and Continuity. Arcs and Curves and tane = y'((t)) x' t 155 Clearly, the values x'(t), y'(t), as well as the d ...

156 Chapter^3 piecewise regular arc the derivatives from the right and from the left must be different from zero. A piecewise sm ...

Functions. Limits and Continuity. Arcs and Curves 157 y y 0 x 0 x (a) (b) Fig. 3.18 6 > O, of the point ti, i.e., N.y(zi) = { ...

158 Chapter3 The following proposition was first stated by C. Jordan in 1892 [10], but his original proof was inadequate. Correc ...

Functions. Limits and Continuity. Arcs and Curves 159 length of such an arc is invariant under a change of parameter t = h( 1), ...

160 Chapter^3 Let 1: z = z(t) 0 S t S 1, be a piecewise smooth arc with a corner point at z 1 = z(t 1 ), 0 < t 1 < 1. Sho ...

Functions. Limits and Continuity. Arcs and Curves 161 3.15 Deformation of Arcs and Curves. Homotopy Definition 3.28 Let 11: [a,b ...

162 Chapter^3 Fig. 3.21 In our later work the following two special types of restricted homotopies will be of particular interes ...

Functions. Limits and Continuity. Arcs and Curves 163 Proof As before, suppose that 11: z = z 1 (t) and 12 : z = z 2 (t), a:::; ...

164 Chapter3 Fig. 3.23 Other alternative definitions of a simply connected set will be given later. 3.16 The Winding Number of a ...