1550251515-Classical_Complex_Analysis__Gonzalez_

154 Chapter^3

where a = h(a'), (J = h((J'), have the same oriented graph. Because of
this, they share some common properties. For instance, they have the
same initial and terminal points, as well as the same length (if they are
rectifiable). Also, the integrals of continuous functions (to be defined later)
along two such arcs of class C^1 (see Definition 3.18) have the same value.
In fact, all such arcs may be considered as forming an equivalence class.
Each of the functions z(t) of the equivalence class is then looked upon as
a specific parametric representation of the arc, and the process of passing
from one parametric representation of the arc to another, by means of a
strictly increasing continuous function h, is called an admissible change
of parametric representation or a reparametrization of the arc (see also
Theorem 3.15).

Definition 3.18 If z'(t) = x'(t) + iy(t) exists for a:::; t:::; (J, the arc I is
said to be differentiable. If z' ( t) exists and is continuous for a :::; t :::; (J the
arc is said to be continuously differentiable or of class C^1. If, in addition,
z^1 ( t) "/:- 0 for a :::; t :::; (J the arc is called regular or smooth.
The graph of a smooth arc has a tangent at every interior point (a
semitangent at the endpoints) with slope dy/dx = y'(t)/x'(t).
If we let

z'(t) = lz'(t)Jeie (3.13-1)

where B = argz'(t), then B is the inclination angle of the tangent to the
graph of the arc at the point z(t) (Fig. 3.16).
In fact, from (3.13-1) we have

so that

y

0

Fig. 3.16

x' (t) + iy'(t) = lz'(t)/( cos B + i sin B)

cos B = x'(t)

/z'(t)I '

x

sine= y'(t)

/z'(t)I