1550251515-Classical_Complex_Analysis__Gonzalez_

Functions. Limits and Continuity. Arcs and Curves 151

However, in what follows we restrict our attention to oriented arcs whose
ranges lie in the complex plane. These are called plane arcs. Symbolically,
we write 1: [a, ,8] ~ <C.
Every oriented arc 'Y will be specified by means of a complex function
of a real variable

z = z(t) = x(t) + iy(t)
defined for a ::::; t ::::; ,8 with continuous components x(t), y(t) over [a, ,8].
We shall indicate briefly this mapping by writing 1: z = z(t), a ::::; t ::::; ,8.
The real variable t is called the parameter of the arc, and the equation
z = z( t) is said to be the parametric equation of the arc. Alternatively, the
mapping may be given by the two real continuous functions

x = x(t), y = y(t)

of the real parameter t defined over [a, ,8].

If the function z = z(t) is a constant, the arc is said to be a point arc.

This special case will be ruled out in what follows. A continuous function
from the open interval (a, ,8) .into the complex plane is called an open arc.
By the interior of a closed arc is meant the open arc obtained by restricting
t to (a, ,8). Unless otherwise specified, whenever we refer to an arc we shall
mean the closed arc.

Definition 3.17 The graph (or, track, trace) of an arc 'Y is defined to be
the range of the function z = z(t), a::::; t::::; ,8, namely, the set of points

gq = 1* = {z: z = z(t),a::::; t::::; ,8}

(see Fig. 3.12).

The arc is regarded as oriented by the parametrization, which means: If

z 1 = z(t 1 ) and z 2 = z(t2), where a::::; ti < tz ::::; ,8, then z1 is considered as

preceding z 2 , denoted z 1 -< z 2. The points zo = z( a) and z* = z(,8) are the

endpoints of the arc, z 0 being the initial point and z* its terminal point.

It is said that the arc has the direction from Zo to z*. In other terms, the

y

I I

zo

I

I

\ \ I I

0 \ I \ x

0 Cl. 13

Fig. 3.12