1550251515-Classical_Complex_Analysis__Gonzalez_

422 Chapter^7

= J f(z)dz+ J f(z)dz

/1 /2

If 71 and 72 are not in juxtaposition, then we define J /1 + /2 f(z) dz to be

J n f(z)dz + J ~ f(z)dz. More generally, for a chain r = E~-1 - ffik"Yk we

define fr f ( z) dz to be

tmk j f(z)dz

k=l lk

3. In view of Definition 3.20, we have

J f(z)dz= 1: f(z(a+f3~t))z'(a+f3-t)(-dt)

-1

By making the change of variable t' = a + (3 - t, which is permissible by

Theorem 7.1-8, we obtain

J f(z)dz =le.. f(z(t'))z'(t')dt' = - 1: f(z(t'))z'(t')dt'

-1

= - j f(z)dz

I

4. Let 'Y be a closed path and suppose first that we have z 0 as the initial
   point, then that we have z~ =/= z 0 as the initial point (this is attained by a
   shift of the parameter). The two points z 0 , z~ divide "Y into two arcs 'Yi and
   72 (with the induced orientation), and the stated property follows from

1 f(z) dz= 1 f(z) dz= 1 f(z) dz

I ~+~ ~+n

5. By using Theorem 7.1 (6) and (7.5-2), we have

1f(z)dz=1: f(z(t))z'(t)dt = J f(z)dz

I I

6. On applying Theorem 7.1 (7), we obtain

J f(z) dz = lip f(z(t))z'(t) dtl

I