56
VIII – Cauchy Theory
given, like the “ function ” ζ = z
1/3 , by an equation P (z, ζ) = 0, where P is
a polynomial (see chap. X).
6 – Dixon’s Theorem
One may wonder in what cases Cauchy’s integral formulas continues to hold
when μ is not null-homotopic in G. Though the proof
38 forms a double-sided
page of ingenious but perfectly elementary arguments, it is was not until 1971
that this was known without recourse to heuristic arguments. To state it, we
need to again use notions defined at the end of n
◦ 4, (i), namely the interior
and exterior of a closed path μ. Hence
C = Ext(μ) ∪ Int(μ) ∪ Supp(μ) ,
these sets being pairwise disjoint and, except the first one, bounded.
Theorem 8 (Dixon). Let G be a domain and μ a closed path in G. The
following statements are equivalent :
(i) The integral along μ of any holomorphic function on G is zero.
(ii) The interior of μ is contained in G.
(iii)
For any holomorphic function on G and any a ∈ G − Supp(μ),
μ
f (z)
z − a
dz = 2πi Ind μ (a)f (a)
(6.1)
The proof consists in showing easy logical implications except for the
second one. We will provide more details for it than its inventor.
(i) =⇒ (ii). For some a /
∈ G, consider the function f (z) = 1/(z − a); by
(i), its integral along μ is zero, but, up to a factor 2πi, it is also the index of
a with respect to μ; the latter is, therefore, zero, proving (ii).
(ii) =⇒ (iii). This is Dixon’s contribution. Define a function g on G × G
by setting
g(ζ, z) = [f (ζ) − f (z)] /(ζ − z) if ζ = z ,
(6.2)
= f
(z) if ζ = z .
By definition of the index, (1) is equivalent to
μ
g(ζ, a)dζ = 0.
(6.1’)
To prove (1’) for a given path μ, we proceed step-by-step by showing that
38 J.D. Dixon, A brief proof of Cauchy’s integral theorem (Proc. Amer. Math. Soc.,
29, 1971, pp. 625–626), reproduced in Remmert, Funktionentheorie 1, Chap. 9,
§ 5, which I follow except for a few details.
VIII – Cauchy Theory
given, like the “ function ” ζ = z
1/3 , by an equation P (z, ζ) = 0, where P is
a polynomial (see chap. X).
6 – Dixon’s Theorem
One may wonder in what cases Cauchy’s integral formulas continues to hold
when μ is not null-homotopic in G. Though the proof
38 forms a double-sided
page of ingenious but perfectly elementary arguments, it is was not until 1971
that this was known without recourse to heuristic arguments. To state it, we
need to again use notions defined at the end of n
◦ 4, (i), namely the interior
and exterior of a closed path μ. Hence
C = Ext(μ) ∪ Int(μ) ∪ Supp(μ) ,
these sets being pairwise disjoint and, except the first one, bounded.
Theorem 8 (Dixon). Let G be a domain and μ a closed path in G. The
following statements are equivalent :
(i) The integral along μ of any holomorphic function on G is zero.
(ii) The interior of μ is contained in G.
(iii)
For any holomorphic function on G and any a ∈ G − Supp(μ),
μ
f (z)
z − a
dz = 2πi Ind μ (a)f (a)
(6.1)
The proof consists in showing easy logical implications except for the
second one. We will provide more details for it than its inventor.
(i) =⇒ (ii). For some a /
∈ G, consider the function f (z) = 1/(z − a); by
(i), its integral along μ is zero, but, up to a factor 2πi, it is also the index of
a with respect to μ; the latter is, therefore, zero, proving (ii).
(ii) =⇒ (iii). This is Dixon’s contribution. Define a function g on G × G
by setting
g(ζ, z) = [f (ζ) − f (z)] /(ζ − z) if ζ = z ,
(6.2)
= f
(z) if ζ = z .
By definition of the index, (1) is equivalent to
μ
g(ζ, a)dζ = 0.
(6.1’)
To prove (1’) for a given path μ, we proceed step-by-step by showing that
38 J.D. Dixon, A brief proof of Cauchy’s integral theorem (Proc. Amer. Math. Soc.,
29, 1971, pp. 625–626), reproduced in Remmert, Funktionentheorie 1, Chap. 9,
§ 5, which I follow except for a few details.