1550251515-Classical_Complex_Analysis__Gonzalez_

Integration 417

The sum Sp is called the variation of z(t) over the interval [a, ,B] corre-

sponding to the partition P, sometimes denoted V!(z, P). The number

sup Sp is called the total variation of the function z(t) in the interval

[a,,B]. If sup Sp< +oo, the function z(t) is said to be of bounded vari-

ation over [a, ,B]. Only in this case the arc 1 is said to be rectifiable.

Hence 1 is rectifiable iff z(t) is a function of bounded variation over

[a, ,B].

Since

and

it is clear that the sum Sp and the corresponding sums

n n

Tp;,, L jx(tk) - x(tk-1)!, Up= L jy(tk) -y(tk-1)1

k=l

are related as follows:

k=l

Tp-5,Sp-5,Tp+Up

Up;::; Sp;::; Tp +Up

Hence if the function z( t) is of bounded variation, so are the functions x( t)
and y(t), and conversely.
The integral of a continuous function along a rectifiable arc will now
be defined.

Definition 7 .6 Let f (z) be a continuous function along the rectifiable arc

1: z = z(t), a ;::; t;::; ,8. Let P ={to, tl, ... , tn} be any partition of [a, ,B],

!Pl= maxk itk -tk-1!,zk = z(tk) and zZ = z(rk), where Tk is an arbitrary

point in [tk-1' tk], k = 1, 2, ... , n (Fig. 7.3).

We note that the distribution of the points z1, z2, ... , Zn-1 on 1*

need not be as simple as in Fig. 7.3. In fact, as t describes the interval
[a,,B] from a to ,8, its image z = z(t) may not move always in the same
direction along 1*. To see this consider, for instance, the mapping defined

by z = sin t + i sin t, 0 ;::; t 5: 37r /2.

The integral of f along 1 is defined by