The Cauchy integral formula is the most important theorem in complex analysis. It will
be established for a disk in this chapter and later will be generalized to much more
general situations but the version given here will suffice to prove many interesting
theorems needed in the later development of the theory. The following are some advanced
calculus results.
Lemma 50.3.1Let f :
[a,b]
→ ℂ. Then f^{′}
(t)
exists if and only ifRef^{′}
(t)
andImf^{′}
(t)
exist. Furthermore,
f′(t) = Ref′(t)+ iIm f′(t).
Proof:The if part of the equivalence is obvious.
Now suppose f^{′}
(t)
exists. Let both t and t + h be contained in
[a,b]
||Re f (t+ h) − Ref (t) || ||f (t+ h)− f (t) ||
||------------------− Re (f′(t))|| ≤ ||------------− f′(t)||
h h
and this converges to zero as h → 0. Therefore, Ref^{′}
(t)
= Re
(f′(t))
. Similarly,
Imf^{′}
(t)
= Im
(f′(t))
.
Lemma 50.3.2If g :
[a,b]
→ ℂ and g is continuous on
[a,b]
and differentiableon
(a,b)
with g^{′}
(t)
= 0, then g
(t)
is a constant.
Proof:From the above lemma, you can apply the mean value theorem to the real
and imaginary parts of g.
Applying the above lemma to the components yields the following lemma.
Lemma 50.3.3If g :
[a,b]
→ ℂ^{n} = X and g is continuous on
[a,b]
anddifferentiable on
(a,b)
with g^{′}
(t)
= 0, then g
(t)
is a constant.
If you want to have X be a complex Banach space, the result is still true.
Lemma 50.3.4If g :
[a,b]
→ X and g is continuous on
[a,b]
and differentiableon
(a,b)
with g^{′}
(t)
= 0, then g
(t)
is a constant.
Proof: Let Λ ∈ X^{′}. Then Λg :
[a,b]
→ ℂ . Therefore, from Lemma 50.3.2, for each
Λ ∈ X^{′},Λg
(s)
= Λg
(t)
and since X^{′} separates the points, it follows g
(s)
= g
(t)
so g is
constant.
Lemma 50.3.5Let ϕ :
[a,b]
×
[c,d]
→ ℝ be continuous and let
∫
b
g(t) ≡ a ϕ(s,t)ds. (50.3.1)
(50.3.1)
Then g is continuous. If
∂∂ϕt
exists and is continuous on
[a,b]
×
[c,d]
, then
∫ b
g′(t) = ∂ϕ(s,t)ds. (50.3.2)
a ∂t
(50.3.2)
Proof: The first claim follows from the uniform continuity of ϕ on
[a,b]
×
[c,d]
,
which uniform continuity results from the set being compact. To establish 50.3.2, let
t and t + h be contained in
[c,d]
and form, using the mean value theorem,
∫ b
g(t+-h)−-g(t) = 1- [ϕ (s,t +h )− ϕ(s,t)]ds
h h ∫a
1- b∂ϕ-(s,t+-θh)
= h a ∂t hds
∫ b∂ϕ (s,t+ θh)
= -----------ds,
a ∂t
where θ may depend on s but is some number between 0 and 1. Then by the uniform
continuity of
This is a very remarkable result. It shows the existence of one continuous derivative
implies the existence of all derivatives, in contrast to the theory of functions of a real
variable. Actually, more than what is stated in the theorem was shown. The above proof
establishes the following corollary.