1550251515-Classical_Complex_Analysis__Gonzalez_

414 Chapter^7

complex function of the real variable t over the interval [a, ,B]. The integral
defined in (7.3-1) is also called a complex line integral.
At times it is convenient to indicate the endpoints z 0 = z( a) and z1 =
z(,B) of the arc 'Y along which the integral is taken. This will be denoted
by writing

1

z1

("Y) f(z)dz

zo
The function f(z) is called the integrand, the arc 'Y the path of integra-
tion, and the endpoints zo and z 1 the limits of the integral. Of course, the
word "limit" is taken here in the usual nontechnical meaning.

Examples 1. Consider the integral f,,(z^2 + 1) dz where "(: z = z(t) =

t + it^2 , 0 $ t $ 1. By (7.3-1) we have

J (z^2 + 1) dz= fo

1

(t^2 + 2it^3 - t^4 + 1)(1+2it) dt

'Y

1 ' 1 '

= 1 (1+t^2 -5t^4 )dt+2i1 (t+2t^3 -t^5 )dt

1+5i

=-3-

2. Consider("'() J;^1 dz, where 'Y: z = z(t), a$ t $ ,8, is any continuously
   differentiable arc co~necting the points z 0 and z 1 • We have, using Theorem
   7.1-9,

1

z1 1(3

('Y) dz= z'(t)dt=z(,B)-z(a)=z1-zo

zo ()( '

Thus the integral depends on the endpoints of the arc but not on the arc
itself.

3. Consider ('Y) J:a^1 z dz along the same arc 'Y as in example 2. We have

{

Zl 1(3

("Y) J zdz = z(t)z'(t) dt = %z(,8)^2 - %z(a)^2 = %(z~ - z3)

zo ()(

Again the integral depends only on the endpoints of the arc. This is a
general property of a class of complex integrals that we shall investigate
later. It is clear that if the terminal point z 1 coincides with the initial
point z 0 , i.e., if 'Y is a closed curve, then

J dz=O

'Y

and (7.3-2)