MH2801: Complex Methods for the Sciences
8
Contour Integration
Contour integration
is a powerful technique, based on complex analysis, that allows us
to solve certain integrals that are otherwise difficult or impossible. Contour integrals have
important applications in many areas of physics, particularly in the study of waves and
oscillations.
8.1
Contour integrals
We have previously studied what it means to take the integral of a real function. To recap:
if
f
(
x
) is a real function, the integral from
x
=
a
to
x
=
b
is defined by dividing the interval
into
N
segments, and evaluating the sum of
f
(
x
)Δ
x
on each segment, in the limit where
N
goes to infinity:
Z
b
a
dx f
(
x
) =
lim
N
→
0
N
X
n
=0
Δ
x f
(
x
n
)
,
where
x
n
=
a
+
n
Δ
x,
Δ
x
=
b

a
N
.
(1)
Now consider the case where
f
is a complex function of a complex variable. The straight
foward way to define the integral of
f
(
z
) is by an analogous expression like this:
lim
N
→
0
N
X
n
=0
Δ
z f
(
z
n
)
(2)
However, since
f
takes complex inputs, the values of
z
n
need not lie along the real line. In
general, the complex numbers
z
n
form a set of points in the twodimensional complex plane.
We can imagine chaining together a sequence of points
z
1
, z
2
, . . . , z
N
, which are separated
by displacements Δ
z
1
,
Δ
z
2
,
Δ
z
3
, . . . ,
Δ
z
N

1
, such that
z
2
=
z
1
+ Δ
z
1
,
z
3
=
z
2
+ Δ
z
2
,
z
4
=
z
3
+ Δ
z
3
,
.
.
. =
.
.
.
z
N
=
z
N

1
+ Δ
z
N

1
.
(3)
Then the sum we are interested in is
N

1
X
n
=1
Δ
z
n
f
(
z
n
) = Δ
z
1
f
(
z
1
) + Δ
z
2
f
(
z
2
) +
· · ·
+ Δ
z
N

1
f
(
z
N

1
)
.
(4)
58
MH2801: Complex Methods for the Sciences
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 Fall '17
 γ, Methods of contour integration