1550251515-Classical_Complex_Analysis__Gonzalez_

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) is the point (0,0), whereas
the initial point under (c) is the point (1, 1).
(3) As stated above, the image of the point t may move in opposite
directions on the graph as t increases from a to (J, unless we impose a
further restriction on the function z(t), namely, to be one-to-one. For
instance, if

z = z ( t) = cos t + i'

the point z(t) moves from A to B and back as t describes the interval

[-^1 / 2 7r, %7r] from -%7r to^1 krr (Fig. 3.15).

Equation (b) in Example 2 may be obtained from that in (a) by the

change of parameter t = >..^2 • However, the functions z = f(t) = t + it2,

0 ::; t ::; 1, and z = f(>..^2 ) = g(>..) = >..^2 + i>..4, 0 ::; >.. ::; 1, are obviously

different, so that we must consider them as defining different arcs.

Whenever a change of parameter t = h( >..) is accomplished by means of

a strictly increasing continuous function h, the two arcs

{

z(t),

z = z(h(>..)),

y

A I B

I

I

I

0 x

0

Fig. 3.15

a ::; t ::; (J

a',::; >.. ::; (J'