1550251515-Classical_Complex_Analysis__Gonzalez_

Functions. Limits and Continuity. Arcs and Curves

and

tane = y'((t))

x' t

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Clearly, the values x'(t), y'(t), as well as the differentials dx = x'(t)dt,

dy = y'(t)dt, are direction numbers for the tangent at the point z(t).

Any smooth arc 'Y is rectifiable and its length, denoted L('Y), is given by

L('Y) = 1(3 Jx^1 (t)^2 + y'(t)^2 dt = 1(3 [z'(t)[ dt (3.13-2)

More generally, an arc is rectifiable iff the function z = z( t) is of bounded
variation. This is discussed in Section 7.6.
The differential of an arc of class C^1 at the point z( t) is defined by the
formula

ds = [z'(t)[ dt (3.13-3)

and it follows that

ds^2 = dx^2 + dy^2 (3.13-4)

An arc is said to be of class GP (p ;:::: 1 an integer) iff the derivatives z' ( t ),

... , z(Pl(t) exist and are continuous on [a, ,B].

Definition 3.19 If z' ( t) exists and is continuous on [a, ,B] except for a

finite number of points of discontinuity of the first kind, the arc is called
piecewise continuously differentiable (or, for short, piecewise differentiable).
If, in addition, z'(t) #- 0, the arc is called piecewise regular or smooth
(Fig. 3.17).
At a point of discontinuity of the first kind of z'(t) (or "corner" point,
as a and bin Fig. 3.17), the function z(t) is of course continuous, but it has
different derivatives from the right and from the left, which are respectively,
equal to the right and left limits of z' ( t) at those points. In the case of a

y

b

a

0 x

Fig. 3.17