2Contour integration and Cauchy's theorem
IB Complex Methods
2.1 Contour and integrals
With real functions, we can just integrate a function, say, from 0 to 1, since there
is just one possible way we can get from 0 to 1 along the real line. However, in
the complex plane, there are many paths we can take to get from a point to
another. Integrating along different paths may produce different results. So we
have to carefully specify our path of integration.
Definition (Curve). A curve γ(t) is a (continuous) map γ : [0, 1] → C.
Definition (Closed curve). A closed curve is a curve γ such that γ(0) = γ(1).
Definition
(Simple curve)
.
A simple curve is one which does not intersect itself,
except at t = 0, 1 in the case of a closed curve.
Definition (Contour). A contour is a piecewise smooth curve.
Everything we do is going to be about contours. We shall, in an abuse of
notation, often use the symbol
γ
to denote both the map and its image, namely
the actual curve in C traversed in a particular direction.
Notation.
The contour
−γ
is the contour
γ
traversed in the opposite direction.
Formally, we say
(−γ)(t) = γ(1 − t).
Given two contours
γ
1
and
γ
2
with
γ
1
(1) =
γ
2
(0),
γ
1
+
γ
2
denotes the two
contours joined end-to-end. Formally,
(γ
1
+ γ
2
)(t) =
(
γ
1
(2t) t <
1
2
γ
2
(2t −1) t ≥
1
2
.
Definition
(Contour integral)
.
The contour integral
R
γ
f
(
z
) d
z
is defined to be
the usual real integral
Z
γ
f(z) dz =
Z
1
0
f(γ(t))γ
0
(t) dt.
Alternatively, and equivalently, dissect [0
,
1] into 0 =
t
0
< t
1
< ··· < t
n
= 1,
and let z
n
= γ(t
n
) for n = 0, ··· , N . We define
δt
n
= t
n+1
− t
n
, δz
n
= z
n+1
− z
n
.
Then
Z
γ
f(z) dz = lim
∆→0
N−1
X
n=0
f(z
n
)δz
n
,
where
∆ = max
n=0,··· ,N−1
δt
n
,
and as ∆ → 0, N → ∞.
All this says is that the integral is what we expect it to be — an infinite sum.
The result of a contour integral between two points in
C
may depend on the
choice of contour.
Example. Consider
I
1
=
Z
γ
1
dz
z
, I
2
=
Z
γ
2
dz
z
,
where the paths are given by
γ
1
γ
2
θ
1−1 0
In both cases, we integrate from
z
=
−
1 to +1 around a unit circle:
γ
1
above,
γ
2
below the real axis. Substitute z = e
iθ
, dz = ie
iθ
dθ. Then we get
I
1
=
Z
0
π
ie
iθ
dθ
e
iθ
= −iπ
I
2
=
Z
0
−π
ie
iθ
dθ
e
iθ
= iπ.
So they can in fact differ.
Elementary properties of the integral
Contour integrals behave as we would expect them to.
Proposition.
(i) We write γ
1
+ γ
2
for the path obtained by joining γ
1
and γ
2
. We have
Z
γ
1
+γ
2
f(z) dz =
Z
γ
1
f(z) dz +
Z
γ
2
f(z) dz
Compare this with the equivalent result on the real line:
Z
c
a
f(x) dx =
Z
b
a
f(x) dx +
Z
c
b
f(x) dx.
(ii) Recall −γ is the path obtained from reversing γ. Then we have
Z
−γ
f(z) dz = −
Z
γ
f(z) dz.
Compare this with the real result
Z
b
a
f(x) dx = −
Z
a
b
f(x) dx.
(iii) If γ is a contour from a to b in C, then
Z
γ
f
0
(z) dz = f(b) −f(a).
This looks innocuous. This is just the fundamental theorem of calculus.
However, there is some subtlety. This requires
f
to be differentiable at
every point on
γ
. In particular, it must not cross a branch cut. For example,
our previous example had
log z
as the antiderivative of
1
z
. However, this
does not imply the integrals along different paths are the same, since we
need to pick different branches of
log
for different paths, and things become
messy.
(iv)
Integration by substitution and by parts work exactly as for integrals on
the real line.
(v) If γ has length L and |f(z)| is bounded by M on γ, then
Z
γ
f(z) dz
≤ LM.
This is since
Z
γ
f(z) dz
≤
Z
γ
|f(z)|| dz| ≤ M
Z
γ
|dz| = ML.
We will be using this result a lot later on.
We will not prove these. Again, if you like proofs, go to IB Complex Analysis.
Integrals on closed contours
If
γ
is a closed contour, then it doesn’t matter where we start from on
γ
;
H
γ
f
(
z
) d
z
means the same thing in any case, so long as we go all the way round
(
H
denotes an integral around a closed contour).
The usual direction of traversal is anticlockwise (the “positive sense”). If
we traverse
γ
in a negative sense (clockwise), then we get negative the previous
result. More technically, the positive sense is the direction that keeps the interior
of the contour on the left. This “more technical” definition might seem pointless
— if you can’t tell what anticlockwise is, then you probably can’t tell which is
the left. However, when we deal with more complicated structures in the future,
it turns out it is easier to define what is “on the left” than “anticlockwise”.
Simply connected domain
Definition
(Simply connected domain)
.
A domain
D
(an open subset of
C
) is
simply connected if it is connected and every closed curve in
D
encloses only
points which are also in D.
In other words, it does not have holes. For example, this is not simply-
connected:
These “holes” need not be big holes like this, but just individual points at which
a function under consider consideration is singular.