In the theory of functions of a complex variable, the most important results are those involving contour
integration. I will use contour integration on curves of bounded variation as in [32], [92], [63] and referred to
in [44]. This is more general than piecewise C^{1} curves but most results can be obtained from only
considering the special case. The most important tools in complex analysis are Cauchy’s theorem in some
form and Cauchy’s formula for an analytic function. This section will give one of the very best versions of
these theorems. They all involve something called a contour integral. Now a contour integral is just a sort
of line integral. As earlier, γ^{∗} will denote the set of points and γ will denote a parametrization. Here is the
definition. It should look familiar and resemble a corresponding definition for line integrals presented
earlier.
Definition 15.3.1Let γ :
[a,b]
→ ℂ be continuous and of bounded variation and let f : γ^{∗}→ Xwhere X is a complete complex normed linear space, usually ℂ. Letting P ≡
{t0,⋅⋅⋅,tn}
wherea = t_{0}< t_{1}<
⋅⋅⋅
< t_{n} = b, define
∥P ∥ ≡ max {|tj − tj−1| : j = 1,⋅⋅⋅,n}
and the Riemann Stieltjes sum by
∑n
S(P ) ≡ f (γ (τj))(γ (tj)− γ (tj−1))
j=1
where τ_{j}∈
[tj−1,tj]
. (Note this notation is a little sloppy because it does not identify the specific point, τ_{j}used. It is understood that this point is arbitrary.) Define∫_{γ}f
(z)
dz as the unique numberwhich satisfies the following condition. For all ε > 0 there exists a δ > 0 such that if
∥P∥
≤ δ,then
|∫ |
|| f (z)dz − S (P)||< ε.
|γ |
Sometimes this is written as
∫
f (z)dz ≡ lim S(P) .
γ ∥P∥→0
You note that this is essentially the same definition given earlier for the line integral only this time
the function has values in ℂ
(more generally X )
rather than ℝ^{n} and there is no dot product
involved. Instead, you multiply by the complex number γ
(tj)
− γ
(tj− 1)
in the Riemann Stieltjes
sum.
Since the contour integral is defined in terms of limits of sums, it follows that the contour integral is
linear because sums are linear. This is just like what was done earlier for line integrals.
The fundamental result in this subject is the following theorem.
Theorem 15.3.2Let f : γ^{∗}→ X be continuous and let γ :
[a,b]
→ ℂ be continuous and ofbounded variation. Then∫_{γ}fdz exists. Also letting δ_{m}> 0 be such that
|t− s|
< δ_{m}implies
∥f (γ(t))− f (γ(s))∥
<
1-
m
,
∥∫ ∥
∥∥ ∥∥ 2V (γ,[a,b])
∥ γfdz − S (P)∥ ≤ m
whenever
∥P∥
< δ_{m}. In addition to this, if γ has a continuous derivative on
[a,b]
, the derivative takenfrom left or right at the endpoints, then
∫ ∫
b ′
γ fdz = a f (γ (t))γ (t)dt (15.7)
(15.7)
Proof: The function, f ∘ γ , is uniformly continuous because it is defined on a compact set.
Therefore, there exists a decreasing sequence of positive numbers,
{δm}
such that if
|s− t|
< δ_{m},
then
1-
∥f (γ(t)) − f (γ(s))∥ < m .
Let
-----------------
Fm ≡ {S (P) : ∥P ∥ < δm }.
Thus F_{m} is a closed set. (The symbol, S
(P )
in the above definition, means to include all sums
corresponding to P for any choice of τ_{j}.) It is shown that
2V (γ,[a,b])
diam (Fm) ≤ ---------- (15.8)
m
(15.8)
and then it will follow there exists a unique point, I ∈∩_{m=1}^{∞}F_{m}. This is because X, the space
where f has its values is complete and Theorem 15.1.2. It will then follow I = ∫_{γ}fdz. To
verify 15.8, it suffices to verify that whenever P and Q are partitions satisfying
∥P∥
< δ_{m} and
∥Q∥
< δ_{m},
∥S (P )− S (Q )∥ ≤-2V (γ,[a,b]). (15.9)
m
(15.9)
Suppose
∥P∥
< δ_{m} and Q ⊇ P. Then also
∥Q ∥
< δ_{m}. To begin with, suppose that P ≡
{t0,⋅⋅⋅,tp,⋅⋅⋅,tn}
and Q ≡
{t0,⋅⋅⋅,tp− 1,t∗,tp,⋅⋅⋅,tn}
. Thus Q contains only one more point than P. Letting S
(Q)
and
S
(P)
be Riemann Steiltjes sums,
p∑−1 ∗
S (Q ) ≡ f (γ (σj))(γ(tj)− γ(tj−1))+ f (γ (σ∗))(γ(t )− γ(tp− 1))
j=1
∑n
+f (γ(σ∗))(γ(tp)− γ(t∗))+ f (γ (σj))(γ(tj) − γ(tj−1)),
j=p+1
p∑−1
S(P) ≡ f (γ(τj))(γ (tj)− γ(tj−1)) +
j=1
=f(γ(τp))(γ(tp)−γ(tp−1))
◜-----------∗---------◞◟-------------------∗-◝
f (γ(τp))(γ(t )− γ(tp− 1))+ f (γ (τp))(γ(tp)− γ(t))
n
+ ∑ f (γ(τj))(γ (tj)− γ(tj−1)).
j=p+1
Therefore,
p−∑ 1
∥S(P) − S (Q)∥ ≤ 1-|γ(tj)− γ (tj−1)|+ 1-|γ (t∗)− γ (tp−1)|+
j=1 m m
1- ∗ n∑ 1- 1-
m |γ(tp)− γ (t )|+ m |γ(tj)− γ (tj−1)| ≤ m V (γ,[a,b]). (15.10)
j=p+1
(15.10)
Clearly the extreme inequalities would be valid in 15.10 if Q had more than one extra point.
You simply do the above trick more than one time. Let S
(P)
and S
(Q)
be Riemann Steiltjes
sums for which
∥P∥
and
∥Q∥
are less than δ_{m} and let R ≡ P ∪ Q. Then from what was just
observed,
∥S (P)− S (Q)∥ ≤ ∥S (P)− S (R)∥+ ∥S (R )− S(Q )∥ ≤-2V (γ,[a,b]).
m
and this shows 15.9 which proves 15.8. Therefore, there exists a unique point, I ∈∩_{m=1}^{∞}F_{m} which
satisfies the definition of ∫_{γ}fdz.
Now consider the case where γ is C^{1}. By uniform continuity, there is δ > 0 such that
| |
||γ′(s)− γ(t)−-γ(s)||≡ f (t,s) < ε (15.11)
| t− s |
(15.11)
if
|t− s|
< δ. To see this, note that f :
[a,b]
×
[a,b]
→ [0,∞) is continuous if the difference quotient is
defined to equal γ^{′}
(s)
when t = s. Since this product is a compact set, it follows that for each ε > 0 there
exists a δ > 0 such that if
|( )|
|t− ˆt,s− ˆs |
< δ, then
| ( )|
|f (t,s)− f ˆt,ˆs|
< ε. So pick
ˆt
= ŝ = s. Then if
|t− s|
< δ,
|f (t,s)|
< ε. Thus
′
|γ (s)(t− s)− (γ(t)− γ(s))| < ε |t− s| (15.12)
(15.12)
Then in the above argument, let
||P ||
m
< min
(η ,δ )
m m
where η_{m} corresponds in the above to ε =
-1
m
.
From the above fundamental estimate,
∥∫ ∥
∥∥ fdz − S(P )∥∥ ≤ 2V-(γ,[a,b])
∥ γ m ∥ m
for any sum
||Pm ||
< δ_{m}. In particular, for such a partition t_{0},
∥∥∫ n ∥∥
∥∥ fdz − ∑ f (γ (t ))γ′(t )(t − t )∥∥
∥∥ γ j=1 j−1 j−1 j j−1 ∥∥
n
≤ 2V (γ,[a,b])+ M ∑ 1-(t − t ) (15.13)
m j=1 m j j−1
= 2V (γ,[a,b])+ M-(b− a) (15.14)
m m
where M is an upper bound for
||f (z)||
for z ∈ γ^{∗}, M existing because of the extreme value theorem.
However, the first part shows as a special case that, since t → f
(γ (t))
γ^{′}
(t)
is continuous, the integral
∫_{a}^{b}f
(γ(t))
γ^{′}
(t)
dt also exists. Let γ
(t)
= t for example. It clearly has bounded variation. Thus, in
particular,
∥∥∑n ∫ b ∥∥
lim ∥∥ f (γ(tj−1))γ′(tj−1)(tj − tj−1)− f (γ (t))γ′(t)dt∥∥ = 0
m→∞ ∥∥j=1 a ∥∥
Therefore, passing to the limit in both sides of 15.14 yields the desired conclusion in 15.7.
■
Observation 15.3.3In the case that γ^{∗} =
[a,b]
an interval on the real line, 15.7shows that ifγ is oriented from a to b, then∫_{γ}f
(z)
dz = ∫_{a}^{b}f
(z)
dz and if γ is oriented from b to a, then∫_{γ}f
(z)
dz = ∫_{b}^{a}f
(z)
dz where the notation on the right signifies the usual Riemann integral.
In the case that f has complex values, it is reasonable to ask for the real and imaginary parts of the
contour integral. It turns out these are just line integrals. Let z = x + iy and let f
(z)
≡ u
(x,y)
+ iv
(x,y)
.
Also let the parametrization be γ
(t)
≡ x
(t)
+ iy
(t)
. Then a term in the approximating sum is of the form
= u (x (ti)− x(ti− 1))− v (y (ti)− y(ti−1))
+i [v (x (ti)− x(ti−1))+ u (y (ti)− y(ti−1))]
Thus the contour integral is really the sum of two line integrals
∫ ∫
(u(x,y),− v (x,y))⋅dr +i (v(x,y),u(x,y))⋅dr
C C
where r
(t)
≡
(x(t),y (t))
. This shows the following simple observation.
Observation 15.3.4When f : γ^{∗}→ ℂ is continuous for γ a bounded variation curve in ℂ, it followsthat
∫ ∫ ∫
fdz = (u(x,y),− v (x,y))⋅dr + i (v(x,y),u(x,y))⋅dr
γ C C
where r
(t)
≡
(Reγ (t),Im γ(t))
,u
(x,y)
≡Ref
(x,y)
, and v
(x,y)
≡Imf
(x,y)
and C is the orientedcurve whose parametrization is r
(t)
.
As in the case of line integrals, these contour integrals are independent of parametrization in the
sense that if γ
(t)
= η
(s)
where t = t
(s)
with s → t
(s)
an increasing continuous function, then
∫_{γ}fdz = ∫_{η}fdw.
Definition 15.3.5If one reverses the order in which points of γ^{∗}are encountered, then one replaces γwith −γ in which, for γ :
[a,b]
→ ℂ, −γ
(t)
encounters the points of γ^{∗}in the opposite order, the definitionof the contour integral shows that
∫ ∫
− fdz = fdz
γ −γ
You could get a parametrization for −γ as −γ
(t)
≡ γ
(b− t)
for t ∈
[0,b− a]
or if you wanted to use thesame interval, define −γ :
[a,b]
→ ℂ by −γ
(t)
≡ γ
(b+ a− t)
.
Now this yields an easy way to check orientation of a rectifiable simple closed curve. Recall that these
simple closed curves are homeomorphisms of S^{1} and there are exactly two orientations.
Theorem 15.3.6Let Γ be a simple closed curve in ℂ and let z ∈ U_{i}, the inside component of Γ^{C}. Thenfor γ a parametrization of Γ,
in real and imaginary parts with z = a + ib, the contour integral is
( )
∫ x− a y− b
------2--------2,------2--------2 ⋅dr
γ (x(− a) + (y− b) (x− a) + (y− b) )
∫ ----− (y-− b)---------x−-a------
+i γ (x − a)2 + (y − b)2,(x − a)2 +(y − b)2 ⋅dr
Using Green’s theorem as in Proposition 12.5.3, we can reduce the integral over Γ to an integral over C_{r}
where C_{r} is a small circle centered at
(a + ib)
. The orientation of the small circle depends on
the orientation of Γ. If this circle has radius r, and if the circle is oriented counter clockwise,
x = a + r cost,y = b + r sint, these two integrals taken over C_{r} reduce to
∫ 2π (r cos t rsint)
---2-, --2-- ⋅(− r sint,rcost)dt
0∫ (r r )
+i 2π − r-sint, rcost ⋅(− rsint,rcost) dt
0 r2 r2
The first is 0 and the second is 2πi. If one takes the opposite orientation of Γ, then the opposite orientation
must also occur for C_{r} and this would involve replacing t with −t in the parameterization. The first
integral would be 0 the same way and the second is −2πi by a simple computation. Thus,
when the point is inside the curve Γ the above integral yields ±1 depending on which way Γ is
oriented.
If a + ib is not on the inside of Γ,nor on Γ. Then you can directly apply Green’s theorem and get
0 in both integrals because the vector field is C^{1} on U_{i}∪ Γ and in each case, Q_{x}− P_{y} = 0.
■
The expression
21πi
∫_{γ}
w1−z-
dw ≡ n
(γ,z)
is called the winding number. It has been discussed here for a
simple closed curve, but it applies to a general curve also. This can be considered later from a different
point of view.
Definition 15.3.7Given Γ a simple closed curve, the orientationis said to be positive if the windingnumber is 1 and negative if the winding number is −1.
The following theorem follows easily from the above definitions and theorem. One can also see from the
definition that something like the triangle inequality will hold. This is contained in the next
theorem.
Theorem 15.3.8Let f be continuous on γ^{∗}having values in a complex Banach space X, writen asf ∈ C