44
VIII – Cauchy Theory
set K ⊂ G. To see this, let us first prove the following result which holds for
far more general spaces :
Lemma. For any open subset G of C, there exist a sequence of compact
subsets K n and open subsets G n such that
G =
K n , K n ⊂ G n ⊂ K n+1 .
Every compact set contained in G is then contained in some K n .
For every z ∈ C, set d(z) = d(z, C − G). As C − G is closed, relation
d(z) = 0 is equivalent to z ∈ C − G, and so G is the set of z ∈ C such that
d(z) > 0. Besides, it is obvious that
|d (z
) − d (z
)| ≤ d (z
, z
) = |z
− z
|
(
∗ )
for all z
and z
. So the function d is continuous. Then define K n by
z ∈ K n ⇐⇒ d(z) ≥ 1/n & |z| ≤ n .
The subsets K n are closed and bounded, and hence compact; they form an
increasing sequence and G is obviously their union. Besides, every a ∈ K n is
in the interior of K n+1 because, by (
∗ ),
d(a, z) ≤ 1/n − 1/(n + 1) =⇒ d(z) ≥ d(a) − d(a, z) ≥ 1/(n + 1) ,
so that K n+1 contains a disc centered at a. The set G n consisting of the
interior points of K n+1 is, therefore, suitable. Finally, if K is a compact
subset of G, it is covered by the open subsets G n , hence by a finite number
of them. So K ⊂ K n for large n, qed.
Having done this, let us return to formula (4) for a holomorphic function on G − S and a closed path μ in G − S, null-homotopic in G. As the
path μ contracts to a point under a homotopy, it describes a compact compact set K ⊂ G contained in one of the open subsets G n of the Lemma. As
G n ∩ S = S n is finite and as μ is null-homotopic in G n , (18) applies to G n
provided only the points a ∈ S n are included in it. But as the result holds for
sufficiently large n, passing to the limit is trivial (in fact, there are finitely
many non-zero residues), qed. In conclusion :
Theorem 5 (Cauchy’s residue formula). Let G be a domain in C, S a
closed discrete subset of G and f a holomorphic function on G
= G − S.
Then
μ
f (ζ)dζ = 2πi
a ∈ S
Ind μ (a) Res(f, a)
(5.5)
for any admissible closed path μ in G
null-homotopic in G.
If G is simply complex, the formula can be applied to every closed path
in G − S. Hence, if the residues of f are all zero, then the integral of f along
any closed path in G
is zero. As a result :
VIII – Cauchy Theory
set K ⊂ G. To see this, let us first prove the following result which holds for
far more general spaces :
Lemma. For any open subset G of C, there exist a sequence of compact
subsets K n and open subsets G n such that
G =
K n , K n ⊂ G n ⊂ K n+1 .
Every compact set contained in G is then contained in some K n .
For every z ∈ C, set d(z) = d(z, C − G). As C − G is closed, relation
d(z) = 0 is equivalent to z ∈ C − G, and so G is the set of z ∈ C such that
d(z) > 0. Besides, it is obvious that
|d (z
) − d (z
)| ≤ d (z
, z
) = |z
− z
|
(
∗ )
for all z
and z
. So the function d is continuous. Then define K n by
z ∈ K n ⇐⇒ d(z) ≥ 1/n & |z| ≤ n .
The subsets K n are closed and bounded, and hence compact; they form an
increasing sequence and G is obviously their union. Besides, every a ∈ K n is
in the interior of K n+1 because, by (
∗ ),
d(a, z) ≤ 1/n − 1/(n + 1) =⇒ d(z) ≥ d(a) − d(a, z) ≥ 1/(n + 1) ,
so that K n+1 contains a disc centered at a. The set G n consisting of the
interior points of K n+1 is, therefore, suitable. Finally, if K is a compact
subset of G, it is covered by the open subsets G n , hence by a finite number
of them. So K ⊂ K n for large n, qed.
Having done this, let us return to formula (4) for a holomorphic function on G − S and a closed path μ in G − S, null-homotopic in G. As the
path μ contracts to a point under a homotopy, it describes a compact compact set K ⊂ G contained in one of the open subsets G n of the Lemma. As
G n ∩ S = S n is finite and as μ is null-homotopic in G n , (18) applies to G n
provided only the points a ∈ S n are included in it. But as the result holds for
sufficiently large n, passing to the limit is trivial (in fact, there are finitely
many non-zero residues), qed. In conclusion :
Theorem 5 (Cauchy’s residue formula). Let G be a domain in C, S a
closed discrete subset of G and f a holomorphic function on G
= G − S.
Then
μ
f (ζ)dζ = 2πi
a ∈ S
Ind μ (a) Res(f, a)
(5.5)
for any admissible closed path μ in G
null-homotopic in G.
If G is simply complex, the formula can be applied to every closed path
in G − S. Hence, if the residues of f are all zero, then the integral of f along
any closed path in G
is zero. As a result :