The arrow on the circle indicating the direction of integration as being clockwise
or counterclockwise as viewed looking down on the z-plane. A simple closed curve
Cis said to be traversed in the positive sense if the direction of integration is in a
counterclockwise direction around the boundary and it is said to be traversed in the
negative sense if the direction of integration is in the clockwise direction around the
boundary. One can write ........
...................................
....................
∫
..............................
C
f(z)dz =− ........
...................................
....................
∫
............................
C
f(z)dz. Observe that by changing the
direction of integration one changes the sign of the integral.
Example 12-20. (Contour integration)
For nan integer, and z 0 ,ρconstants, integrate the function f(z) = (z−z 0 )naround
a circle of radius ρcentered at the point z 0. Perform the integration in the positive
sense.
Solution: Let Cdenote the circle |z−z 0 |=ρof radius ρcentered at the point z 0. The
curve Ccan be represented in the parametric form
z=z(t) = z 0 +ρei t, 0 ≤t≤ 2 π with dz =ρe i tidt.
We then have
...............................................................
∫
............................
C
f(z)dz = .........
.................................
.....................
∫
............................
C
(z−z 0 )ndz =
∫ 2 π
0
ρnei ntiρei tdt =iρn+1
∫ 2 π
0
ei(n+1)tdt.
For ndifferent from − 1 we have .........
.................................
.....................
∫
............................
C
(z−z 0 )ndz =iρ n+1
[
ei(n+1)t
i(n+ 1)
] 2 π
0
= 0 .For nequal
to − 1 we have ...........
.............................
.......................
∫
............................
C
(z−z 0 )ndz = ...........
.............................
.......................
∫
............................
C
dz
z−z 0
=i
∫ 2 π
0
dt = 2πi. Hence, the line or contour
integral of the function f(z) = (z−z 0 )n, with nan integer, which is taken around a
circle Ccentered at z 0 with radius ρ, can be expressed
...............................................................
∫
............................
C
(z−z 0 )ndz =
{ 2 πi when n=− 1
0 when n=− 1 and an integer.
(12.191)
This result will be used quite frequently throughout the remainder of this text.
The Laurent series
For z=x+iy a complex variable and z 0 =x 0 +iy 0 a fixed point in the complex
z-plane, one must deal with the following quantities. (i) The magnitude of zdenoted
