1550251515-Classical_Complex_Analysis__Gonzalez_

160 Chapter^3

9. 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. Show that 1': z = z(h(>.)), 0 SAS 1, where

t = h(.A) = (.A - c)

3

+ c

3

, t1 = h(c)

(1 - c)^3 + c^3

is a smooth reparametrization of 'Y with no corner point at "( = c (Staib

(16]).

3.14 Chains and Cycles

Consider any three arcs 11 ,1 2 ,1 3 , either open or closed, as in Fig. 3.19.

In general, 'YI, 12 , 73 taken together do not give an arc in the usual sense,

since the union of their graphs will not be connected (except in very special

cases). It is convenient, however, to have a name and a notation for such

formal sums as "fl + 12 + 13 , 11 - 272 + 513 , and the like. To this end the
following definitions are introduced.

Definition 3.26 Let 11 , 12 , ••• , l'n be a finite number of oriented arcs,
and let mi, m 2 , .•• , mn be any integers (positive, negative, or zero). A
formal sum of the form
n
r = m111 + m212 + ... + mn1n = .L: mni
i=l

is called a chain. If the graph of every 'Yi lies in an open set A, we say that r

is a chain in A. If the arcs 'Yi are all smooth arcs, the chain is called regular.

Definition 3.27 A chain is a closed chain or a cycle if each point is an
initial point of just as many of the arcs 'Yi as it is a terminal point.
A closed contour is a special case of a cycle.

0 x

Fig. 3.19