§ 2. Cauchy’s Integral Formulas
43
terms indexed by b = a are holomorphic with respect to a, the polar part of
the Laurent series of g with respect to a is obtained by removing from the
series of f that of g a (z) + ρ a /(z − a), i.e. the sum of terms of degree < 0
of the Laurent series of f . Therefore, in reality, the Laurent series of g with
respect to a is a power series, and denoting the constant term of the Laurent
series of f with respect to a by g(a), g is transformed into a function defined
and holomorphic on all of G.
To compute the integral of f along a closed path μ in G
, it then suffices
to compute those of the functions g a , the functions 1/(z − a) and of g.
By definition of the index, to begin with we have
μ
dζ
ζ − a
= 2πi. Ind μ (a) .
(5.3)
As, on the other hand, g is holomorphic on all of G and as by assumption, μ
is null-homotopic, its integral along μ is zero (Corollary 3 of Theorem 3). As
for the function g a , for ζ = a, it is represented by a series
g a (ζ) =
n≤−2
c n (ζ − a)
n
converging in C − {a} and without any terms of degree −1; it, therefore,
admits a primitive
n≤−2
c n (ζ − a)
n+1 /(n + 1)
on C − {a} (Chap. VII, n
◦ 16 : possibility of differentiating a Laurent series
term by term). So irrespective of whether μ is null-homotopic or not, the
integral of g a along μ is zero.
The only terms in sum (2) contributing effectively to the computation of
the integral are, therefore, the fractions ρ a /(ζ − a). Hence, in view of (3), the
relation
μ
f (ζ)dζ = 2πi
a ∈ S
Ind μ (a) Res(f, a)
(5.4)
follows.
This result assumes that f has finitely many singular points in G. In fact,
it remains valid when S is a possibly infinite closed and discrete subset of
the topological space G or, equivalently, has a neighbourhood V for all z ∈ G
such that V ∩ S is finite, or else is such that K ∩ S is finite for any compact
30
30 “ closed in G ” means that every point of G (and not of C) is a limit point of S
and is in S; “ discrete in G ” means that every z ∈ S has a neighbourhood V such
that V ∩ S = {z}. Supposing that K ⊂ G is compact, if K ∩ S is infinite, then
there exists (use BL) a ∈ K such that V ∩S is infinite for every neighbourhood V
of a. Hence, we get a sequence of pairwise distinct points of S converging to a;
since S is closed in G, it follows that a ∈ S, which contradicts the discreteness
assumption on S.
43
terms indexed by b = a are holomorphic with respect to a, the polar part of
the Laurent series of g with respect to a is obtained by removing from the
series of f that of g a (z) + ρ a /(z − a), i.e. the sum of terms of degree < 0
of the Laurent series of f . Therefore, in reality, the Laurent series of g with
respect to a is a power series, and denoting the constant term of the Laurent
series of f with respect to a by g(a), g is transformed into a function defined
and holomorphic on all of G.
To compute the integral of f along a closed path μ in G
, it then suffices
to compute those of the functions g a , the functions 1/(z − a) and of g.
By definition of the index, to begin with we have
μ
dζ
ζ − a
= 2πi. Ind μ (a) .
(5.3)
As, on the other hand, g is holomorphic on all of G and as by assumption, μ
is null-homotopic, its integral along μ is zero (Corollary 3 of Theorem 3). As
for the function g a , for ζ = a, it is represented by a series
g a (ζ) =
n≤−2
c n (ζ − a)
n
converging in C − {a} and without any terms of degree −1; it, therefore,
admits a primitive
n≤−2
c n (ζ − a)
n+1 /(n + 1)
on C − {a} (Chap. VII, n
◦ 16 : possibility of differentiating a Laurent series
term by term). So irrespective of whether μ is null-homotopic or not, the
integral of g a along μ is zero.
The only terms in sum (2) contributing effectively to the computation of
the integral are, therefore, the fractions ρ a /(ζ − a). Hence, in view of (3), the
relation
μ
f (ζ)dζ = 2πi
a ∈ S
Ind μ (a) Res(f, a)
(5.4)
follows.
This result assumes that f has finitely many singular points in G. In fact,
it remains valid when S is a possibly infinite closed and discrete subset of
the topological space G or, equivalently, has a neighbourhood V for all z ∈ G
such that V ∩ S is finite, or else is such that K ∩ S is finite for any compact
30
30 “ closed in G ” means that every point of G (and not of C) is a limit point of S
and is in S; “ discrete in G ” means that every z ∈ S has a neighbourhood V such
that V ∩ S = {z}. Supposing that K ⊂ G is compact, if K ∩ S is infinite, then
there exists (use BL) a ∈ K such that V ∩S is infinite for every neighbourhood V
of a. Hence, we get a sequence of pairwise distinct points of S converging to a;
since S is closed in G, it follows that a ∈ S, which contradicts the discreteness
assumption on S.