and consequently,
∫
C
f(z)dz =
∫
C
(u+iv )(dx +idy )
=
∫
C
(u dx −v dy ) + i
∫
C
(v dx +u dy)
=
∫t 2
t 1
∂U
∂x
x′(t)dt +∂U
∂y
y′(t)dt +i
∫t 2
t 1
∂V
∂x
x′(t)dt +∂V
∂y
y′(t)dt
=
∫z(t 2 )
z(t 1 )
dU +i
∫z(t 2 )
z(t 1 )
dV =
∫z(t 2 )
z(t 1 )
dF =F(z)
z(t 2 )
z(t 1 )
=F(z(t 2 )) −F(z(t 1 ))
Example 12-19. Evaluate the integral I=
∫
C
(z−z 0 )ndz where nis an integer
and C is the arc of the circle defined by z=z(t) = z 0 +r ei t for t 1 ≤t≤t 2 and r > 0
constant. For n=− 1 we have
I=
∫z(t 2 )
z(t 1 )
(z−z 0 )ndz =(z−z^0 )
n+1
n+ 1
z(t 2 )
z(t 1 )
n=− 1
For n=− 1 , substitute z−z 0 =r e i t with dz =r e i t idt to obtain
I=
∫
C
dz
z−z 0 =
∫t 2
t 1
r e i t idt
r e i t =
∫t 2
t 1
idt =i(t 2 −t 1 )
Note that a special case z(t 1 ) = z(t 2 )occurs when the arc of the circle closes on itself.
Closed curves
A plane curve C in the z-plane can be defined by a set of para-
metric equations {x(t), y (t)}for the parameter tranging over a set
of values t 1 ≤t≤t 2 .A curve Cis called a closed curve if the end
points coincide. If C is a closed curve, then the end conditions
satisfy x(t 1 ) = x(t 2 ) and y(t 1 ) = y(t 2 ). If (x 0 , y 0 ) is a point on the
curve C, which is not an end point, and there exists more than one
value of the parameter tsuch that {x(t), y (t)}={x 0 , y 0 },then the
point (x 0 , y 0 )is called a multiple point or point where the curve C
crosses itself. A curve Cis called a simple closed curve if the end
points meet and it has no multiple points.
Whenever a curve Cis a simple closed curve, the line integral of f(z)around C
or contour integral around the curve C is represented by an integral having one of
the forms
...............................................................
∫
............................
C
f(z)dz. or ..........
...............................
......................
∫
..............................
C
f(z)dz (12.190)
