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Chapter 5
Consequences of Cauchy’s Theorem
If things are nice there is probably a good reason why they are nice: and if you do not know at
least one reason for this good fortune, then you still have work to do.
Richard Askey
5.1
Extensions of Cauchy’s Formula
We now derive formulas for
f
°
and
f
°°
which resemble Cauchy’s formula (Theorem 4.10).
Theorem 5.1.
Suppose
f
is holomorphic on the region
G
,
w
∈
G
,and
γ
is a positively oriented,
simple, closed, smooth,
G
contractible curve such that
w
is inside
γ
. Then
f
°
(
w
)=
1
2
πi
°
γ
f
(
z
)
(
z
−
w
)
2
dz
and
f
°°
(
w
)=
1
πi
°
γ
f
(
z
)
(
z
−
w
)
3
dz .
This innocentlooking theorem has a very powerful consequence: just from knowing that
f
is holomorphic we know of the existence of
f
°°
, that is,
f
°
is also holomorphic in
G
. Repeating
this argument for
f
°
, then for
f
°°
,
f
°°°
, etc., gives the following statement, which has no analog
whatsoever in the reals.
Corollary 5.2.
If
f
is diFerentiable in the region
G
then
f
is in±nitely diFerentiable in
G
.
Proof of Theorem 1.1.
The idea of the proof is very similar to the proof of Cauchy’s integral formula
(Theorem 4.10). We will study the following diFerence quotient, which we can rewrite as follows
by Theorem 4.10.
f
(
w
+∆
w
)
−
f
(
w
)
∆
w
=
1
∆
w
±
1
2
πi
°
γ
f
(
z
)
z
−
(
w
+∆
w
)
dz
−
1
2
πi
°
γ
f
(
z
)
z
−
w
dz
²
=
1
2
πi
°
γ
f
(
z
)
(
z
−
w
−
∆
w
)(
z
−
w
)
dz .
53
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View Full DocumentCHAPTER 5. CONSEQUENCES OF CAUCHY’S THEOREM
54
Hence we will have to show that the following expression gets arbitrarily small as ∆
w
→
0:
f
(
w
+∆
w
)
−
f
(
w
)
∆
w
−
1
2
πi
°
γ
f
(
z
)
(
z
−
w
)
2
dz
=
1
2
πi
°
γ
f
(
z
)
(
z
−
w
−
∆
w
)(
z
−
w
)
−
f
(
z
)
(
z
−
w
)
2
dz
=∆
w
1
2
πi
°
γ
f
(
z
)
(
z
−
w
−
∆
w
)(
z
−
w
)
2
dz .
This can be made arbitrarily small if we can show that the integral stays bounded as ∆
w
→
0.
In fact, by Proposition 4.4(d), it suﬃces to show that the
integrand
stays bounded as ∆
w
→
0
(because
γ
and hence length(
γ
) are Fxed). Let
M
= max
z
∈
γ

f
(
z
)

and
N
= max
z
∈
γ

z
−
w

.S
ince
γ
is a closed set, there is some positive
δ
so that the open disk of radius
δ
around
w
does not
intersect
γ
; that is,

z
−
w
≥
δ
for all
z
on
γ
. By the reverse triangle inequality we have for all
z
∈
γ
±
±
±
±
f
(
z
)
(
z
−
w
−
∆
w
)(
z
−
w
)
2
±
±
±
±
≤

f
(
z
)

(

z
−
w
−
∆
w

)

z
−
w

2
≤
M
(
δ
−
∆
w

)
N
2
,
which certainly stays bounded as ∆
w
→
0. The proof of the formula for
f
°°
is very similar and will
be left for the exercises (see Exercise 2).
Remarks.
1. Theorem 1.1 suggests that there are similar formulas for the higher derivatives of
f
.
This is in fact true, and theoretically one could obtain them one by one with the methods of the
proof of Theorem 1.1. However, once we start studying power series for holomorphic functions,
we will obtain such a result much more easily; so we save the derivation of formulas for higher
derivatives of
f
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