In the theory of functions of a complex variable, the most important results are those
involving contour integration. I will base this on the notion of Riemann Stieltjes
integrals as in [?], [?], and [?]. The Riemann Stieltjes integral is a generalization of
the usual Riemann integral and requires the concept of a function of bounded
variation.
Definition 49.0.1Let γ :
[a,b]
→ ℂ be a function. Then γ is of bounded variationif
{ ∑n }
sup |γ (ti)− γ(ti−1)| : a = t0 < ⋅⋅⋅ < tn = b ≡ V (γ,[a,b]) < ∞
i=1
where the sums are taken over all possible lists,
{a = t0 < ⋅⋅⋅ < tn = b}
. The set ofpoints γ
([a,b])
will also be denoted by γ^{∗}.
The idea is that it makes sense to talk of the length of the curve γ
([a,b])
, defined as
V
(γ,[a,b])
. For this reason, in the case that γ is continuous, such an image of a bounded
variation function is called a rectifiable curve.
Definition 49.0.2Let γ :
[a,b]
→ ℂ be of bounded variation and let f : γ^{∗}→ X. LettingP≡
{t0,⋅⋅⋅,tn}
where a = t_{0}< t_{1}<
⋅⋅⋅
< t_{n} = b, define
||P || ≡ max {|t − t | : j = 1,⋅⋅⋅,n}
j j−1
and the Riemann Steiltjes 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 thespecific point, τ_{j}used. It is understood that this point is arbitrary.) Define∫_{γ}fdγ as theunique number which satisfies the following condition. For all ε > 0 there exists a δ > 0
such that if
||P||
≤ δ, then
||∫ ||
|| f dγ − S (P )|| < ε.
γ
Sometimes this is written as
∫
fdγ ≡ lim S (P).
γ ||P||→0
The set of points in the curve, γ
([a,b])
will be denoted sometimes by γ^{∗}.
Then γ^{∗} is a set of points in ℂ and as t moves from a to b, γ
(t)
moves from γ
(a)
to γ
(b)
. Thus γ^{∗} has a first point and a last point. If ϕ :
[c,d]
→
[a,b]
is a
continuous nondecreasing function, then γ ∘ ϕ :
[c,d]
→ ℂ is also of bounded
variation and yields the same set of points in ℂ with the same first and last
points.
Theorem 49.0.3Let ϕ and γ be as just described. Then assuming that
∫
fdγ
γ
exists, so does
∫
γ∘ϕfd (γ ∘ϕ)
and
∫ ∫
fdγ = fd(γ ∘ϕ). (49.0.1)
γ γ∘ϕ
(49.0.1)
Proof:There exists δ > 0 such that if P is a partition of
[a,b]
such that
||P ||
< δ,
then
|∫ |
|| ||
| γfdγ − S(P )| < ε.
By continuity of ϕ, there exists σ > 0 such that if Q is a partition of
This theorem shows that ∫_{γ}fdγ is independent of the particular γ used in its
computation to the extent that if ϕ is any nondecreasing continuous function from
another interval,
[c,d]
, mapping to
[a,b]
, then the same value is obtained by replacing γ
with γ ∘ ϕ.
The fundamental result in this subject is the following theorem. We have in mind
functions which have values in ℂ but there is no change if the functions have values in
any complete normed vector space.
Theorem 49.0.4Let f : γ^{∗}→ X be continuous and let γ :
[a,b]
→ ℂ be continuous andof bounded variation. Then∫_{γ}fdγ 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])
| γfdγ − S (P)| ≤ m
whenever
||P||
< δ_{m}.
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
|f (γ(t))− f (γ(s))| < 1-.
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
diam (F ) ≤ 2V-(γ,[a,b]) (49.0.2)
m m
(49.0.2)
and then it will follow there exists a unique point, I ∈∩_{m=1}^{∞}F_{m}. This is
because X is complete. It will then follow I = ∫_{γ}f
(t)
dγ
(t)
. To verify 49.0.2, it
suffices to verify that whenever P and Q are partitions satisfying
||P||
< δ_{m} and
||Q||
< δ_{m},
2-
|S (P)− S (Q )| ≤ m V (γ,[a,b]). (49.0.3)
(49.0.3)
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(◞t◟p)−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∑−1-1 -1 ∗
|S(P)− S (Q)| ≤ m |γ (tj)− γ(tj− 1)|+ m |γ(t )− γ(tp− 1)|+
j=1
1 ∑n 1 1
--|γ (tp)− γ (t∗)|+ --|γ (tj)− γ (tj−1)| ≤ --V (γ,[a,b]). (49.0.4)
m j=p+1 m m
(49.0.4)
Clearly the extreme inequalities would be valid in 49.0.4 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 )| ≤ 2-V (γ,[a,b]).
m
and this shows 49.0.3 which proves 49.0.2. Therefore, there exists a unique complex
number, I ∈∩_{m=1}^{∞}F_{m} which satisfies the definition of ∫_{γ}fdγ. This proves the
theorem.
The following theorem follows easily from the above definitions and theorem.
Theorem 49.0.5Let f ∈ C
(γ∗)
and let γ :
[a,b]
→ ℂ be of bounded variation andcontinuous. Let
M ≥ max {||f ∘ γ(t)|| : t ∈ [a,b]}. (49.0.5)
(49.0.5)
Then
||||∫ ||||
|||| fdγ|||| ≤ M V (γ,[a,b]). (49.0.6)
γ
(49.0.6)
Also if
{fn}
is a sequence of functions of C
∗
(γ )
which is converging uniformly to thefunction, f on γ^{∗}, then
∫ ∫
lnim→∞ fndγ = fdγ. (49.0.7)
γ γ
(49.0.7)
Proof:Let 49.0.5 hold. From the proof of the above theorem, when
→ ℂ be continuous and of bounded variation. Let Ω
be an open set containing γ^{∗}and let f : Ω × K → X be continuous for K acompact set in ℂ, and let ε > 0 be given. Then there exists η :
[a,b]
→ ℂ such thatη
(a)
= γ
(a)
,γ
(b)
= η
(b)
, η ∈ C^{1}
([a,b])
, and
||γ − η|| < ε, (49.0.8)
(49.0.8)
|∫ ∫ |
|| f (⋅,z)dγ − f (⋅,z)dη||< ε, (49.0.9)
| γ η |
(49.0.9)
V (η,[a,b]) ≤ V (γ,[a,b]), (49.0.10)
(49.0.10)
where
||γ − η||
≡ max
{|γ(t)− η(t)| : t ∈ [a,b]}
.
Proof: Extend γ to be defined on all ℝ according to γ