Complex integrals

24.8 COMPLEX INTEGRALS
w3
y
w3
s
w = g(z)
x1
x2
−1
1
φ1
w1
−a
x
φ2
w2
a
r
Figure 24.7 Transforming the upper half of the z-plane into the interior of
the region −a < r < a, s > 0 in the w-plane.
24.8 Complex integrals
Corresponding to integration with respect to a real variable, it is possible to
define integration with respect to a complex variable between two complex limits.
Since the z-plane is two-dimensional there is clearly greater freedom and hence
ambiguity in what is meant by a complex integral. If a complex function f(z) is
single-valued and continuous in some region R in the complex plane, then we can
define the complex integral of f(z) between two points A and B along some curve
in R; its value will depend, in general, upon the path taken between A and B (see
figure 24.8). However, we will find that for some paths that are different but bear
a particular relationship to each other the value of the integral does not depend
upon which of the paths is adopted.
Let a particular path C be described by a continuous (real) parameter t
(α ≤ t ≤ β) that gives successive positions on C by means of the equations
x = x(t),
y = y(t),
(24.32)
with t = α and t = β corresponding to the points A and B, respectively. Then the
integral along path C of a continuous function f(z) is written
f(z) dz
(24.33)
C
and can be given explicitly as a sum of real integrals as follows:
f(z) dz = (u + iv)(dx + idy)
C
C
=
u dx −
v dy + i u dy + i v dx
C
β
=
α
C
dx
dt −
u
dt
α
C
β
dy
dt + i
v
dt
α
C
β
dy
dt + i
u
dt
β
v
α
dx
dt.
dt
(24.34)
845
COMPLEX VARIABLES
y
B
C2
C1
x
A
C3
Figure 24.8 Some alternative paths for the integral of a function f(z) between
A and B.
The question of when such an integral exists will not be pursued, except to state
that a sufficient condition is that dx/dt and dy/dt are continuous.
Evaluate the complex integral of f(z) = z −1 along the circle |z| = R, starting and finishing
at z = R.
The path C1 is parameterised as follows (figure 24.9(a)):
z(t) = R cos t + iR sin t,
0 ≤ t ≤ 2π,
whilst f(z) is given by
f(z) =
x − iy
1
.
= 2
x + iy
x + y2
Thus the real and imaginary parts of f(z) are
u=
R cos t
x
=
x2 + y 2
R2
and
v=
−y
R sin t
=−
.
x2 + y 2
R2
Hence, using expression (24.34),
2π
2π 1
cos t
− sin t
R cos t dt
dz =
(−R sin t) dt −
R
R
C1 z
0
0
2π
2π cos t
− sin t
(−R sin t) dt
+i
R cos t dt + i
R
R
0
0
= 0 + 0 + iπ + iπ = 2πi. (24.35)
With a bit of experience, the reader may be able to evaluate integrals like
the LHS of (24.35) directly without having to write them as four separate real
integrals. In the present case,
2π
2π
dz
−R sin t + iR cos t
=
dt =
i dt = 2πi.
(24.36)
R cos t + iR sin t
C1 z
0
0
846
24.8 COMPLEX INTEGRALS
y
y
y
C1
C2
iR
C3b
R
R
t
t
x
−R
R x
(a)
C3a
s=1
t=0
−R
(b)
R x
(c)
Figure 24.9 Different paths for an integral of f(z) = z −1 . See the text for
details.
This very important result will be used many times later, and the following should
be carefully noted: (i) its value, (ii) that this value is independent of R.
In the above example the contour was closed, and so it began and ended at
the same point in the Argand diagram. We can evaluate complex integrals along
open paths in a similar way.
Evaluate the complex integral of f(z) = z −1 along the following paths (see figure 24.9):
(i) the contour C2 consisting of the semicircle |z| = R in the half-plane y ≥ 0,
(ii) the contour C3 made up of the two straight lines C3a and C3b .
(i) This is just as in the previous example, except that now 0 ≤ t ≤ π. With this change,
we have from (24.35) or (24.36) that
C2
dz
= πi.
z
(24.37)
(ii) The straight lines that make up the countour C3 may be parameterised as follows:
C3a ,
C3b ,
z = (1 − t)R + itR
for 0 ≤ t ≤ 1;
z = −sR + i(1 − s)R
for 0 ≤ s ≤ 1.
With these parameterisations the required integrals may be written
C3
dz
=
z
1
0
−R + iR
dt +
R + t(−R + iR)
1
0
−R − iR
ds.
iR + s(−R − iR)
(24.38)
If we could take over from real-variable theory that, for real t, (a+bt)−1 dt = b−1 ln(a+bt)
even if a and b are complex, then these integrals could be evaluated immediately. However,
to do this would be presuming to some extent what we wish to show, and so the evaluation
847
COMPLEX VARIABLES
must be made in terms of entirely real integrals. For example, the first is given by
1
1
−R + iR
(−1 + i)(1 − t − it)
dt
dt =
(1 − t)2 + t2
0 R(1 − t) + itR
0
1
1
2t − 1
1
dt + i
dt
=
2
2
0 1 − 2t + 2t
0 1 − 2t + 2t
1
1
t − 12
i
1
2
−1
ln(1 − 2t + 2t ) +
=
2 tan
1
2
2
0
2
0
πi
i π
π
= .
=0+
− −
2 2
2
2
The second integral on the RHS of (24.38) can also be shown to have the value πi/2. Thus
dz
= πi. C3 z
Considering the results of the preceding two examples, which have common
integrands and limits, some interesting observations are possible. Firstly, the two
integrals from z = R to z = −R, along C2 and C3 , respectively, have the same
value, even though the paths taken are different. It also follows that if we took a
closed path C4 , given by C2 from R to −R and C3 traversed backwards from −R
to R, then the integral round C4 of z −1 would be zero (both parts contributing
equal and opposite amounts). This is to be compared with result (24.36), in which
closed path C1 , beginning and ending at the same place as C4 , yields a value 2πi.
It is not true, however, that the integrals along the paths C2 and C3 are equal
for any function f(z), or, indeed, that their values are independent of R in general.
Evaluate the complex integral of f(z) = Re z along the paths C1 , C2 and C3 shown in
figure 24.9.
(i) If we take f(z) = Re z and the contour C1 then
2π
Re z dz =
R cos t(−R sin t + iR cos t) dt = iπR 2 .
C1
0
(ii) Using C2 as the contour,
Re z dz =
C2
π
0
R cos t(−R sin t + iR cos t) dt = 12 iπR 2 .
(iii) Finally the integral along C3 = C3a + C3b is given by
1
1
Re z dz =
(1 − t)R(−R + iR) dt +
(−sR)(−R − iR) ds
C3
0
0
= 12 R 2 (−1 + i) + 12 R 2 (1 + i) = iR 2 . The results of this section demonstrate that the value of an integral between the
same two points may depend upon the path that is taken between them but, at
the same time, suggest that, under some circumstances, the value is independent
of the path. The general situation is summarised in the result of the next section,
848