32
VIII – Cauchy Theory
μ(t) − a = |μ(t) − a| . exp [i.A(t)]
(4.4)
for all t. Assuming μ to be closed, the term log |μ(t) − a| of (3) has the same
values at t = 0 and t = 1 since it only depends on μ(t). Its variation along μ
is, therefore, zero, so that, up to a factor i, that of L(1) − L(0) is equal to the
variation A(1) − A(0) of the argument of μ(t) − a. But as the various possible
values of the argument of a complex number differ by multiples of 2π,
A(1) − A(0) = 2π. Ind μ (a) ,
(4.5)
where Ind μ (a) is an integer called the index of a with respect to μ, unless it
is the index of μ with respect to a. As explained at the end of Chap. IV, § 4,
physically, it is the number of rotations carried out by the ray with initial
point a and passing through μ(t) as t varies from 0 to 1. It is a positive
or negative number calculated by taking into account the direction of the
rotations. This will be justified in Chap. X, no 3, (iii). Clearly, Ind μ (a) only
depends on the homotopy class of μ in C − {a}. As a result, setting
Supp(μ) = μ(I)
for the support of the path μ, the next result follows :
Corollary. For any closed path μ in C and all a ∈ C − Supp(μ),
μ
dz
z − a
=
1
0
μ
(t)
μ(t) − a
dt = 2πi. Ind μ (a)z, .
(4.6)
Note that the left hand side of (6) – and hence the right hand side – is a
continuous function of a outside the compact set μ(I) = Supp(μ), i.e. outside
the image of I under μ, since the function of the (t, a) integrated over [0, 1]
is continuous (Chap. V, no 9, Theorem 9). From the fact that a → Ind μ (a)
has values in Z, it can be deduced that the index of a point a with respect
to a closed path μ only depends on the connected component of a in the open
set C − Supp(μ). By (6), it clearly approaches 0 as |a| increases indefinitely.
Hence it is zero in the unbounded connected component of C − Supp(μ). The
latter is unique because it at least contains the exterior of any disc D having
Supp(μ) as a subset, so that all other components are contained in D, and
hence are bounded.
We sometimes write Ext(μ), exterior of μ, for the set of z ∈ C − Supp(μ)
where Ind μ (z) = 0 and Int(μ), interior of μ, for the set of z where
Ind μ (z) = 0. The exterior of μ contains the non-compact connected component of C − Supp(μ), but may be strictly larger. These notions are not at
all related to those defined in Chap. III, no 1 with respect to an arbitrary
subset of C, but refer instead to what has been said at the end of Chap. III,
§ 4 (Jordan curve theorem) in the case of a “ simple ” curve, i.e. homeomorphic to the unit circle T. This supposedly simple case being already quite
VIII – Cauchy Theory
μ(t) − a = |μ(t) − a| . exp [i.A(t)]
(4.4)
for all t. Assuming μ to be closed, the term log |μ(t) − a| of (3) has the same
values at t = 0 and t = 1 since it only depends on μ(t). Its variation along μ
is, therefore, zero, so that, up to a factor i, that of L(1) − L(0) is equal to the
variation A(1) − A(0) of the argument of μ(t) − a. But as the various possible
values of the argument of a complex number differ by multiples of 2π,
A(1) − A(0) = 2π. Ind μ (a) ,
(4.5)
where Ind μ (a) is an integer called the index of a with respect to μ, unless it
is the index of μ with respect to a. As explained at the end of Chap. IV, § 4,
physically, it is the number of rotations carried out by the ray with initial
point a and passing through μ(t) as t varies from 0 to 1. It is a positive
or negative number calculated by taking into account the direction of the
rotations. This will be justified in Chap. X, no 3, (iii). Clearly, Ind μ (a) only
depends on the homotopy class of μ in C − {a}. As a result, setting
Supp(μ) = μ(I)
for the support of the path μ, the next result follows :
Corollary. For any closed path μ in C and all a ∈ C − Supp(μ),
μ
dz
z − a
=
1
0
μ
(t)
μ(t) − a
dt = 2πi. Ind μ (a)z, .
(4.6)
Note that the left hand side of (6) – and hence the right hand side – is a
continuous function of a outside the compact set μ(I) = Supp(μ), i.e. outside
the image of I under μ, since the function of the (t, a) integrated over [0, 1]
is continuous (Chap. V, no 9, Theorem 9). From the fact that a → Ind μ (a)
has values in Z, it can be deduced that the index of a point a with respect
to a closed path μ only depends on the connected component of a in the open
set C − Supp(μ). By (6), it clearly approaches 0 as |a| increases indefinitely.
Hence it is zero in the unbounded connected component of C − Supp(μ). The
latter is unique because it at least contains the exterior of any disc D having
Supp(μ) as a subset, so that all other components are contained in D, and
hence are bounded.
We sometimes write Ext(μ), exterior of μ, for the set of z ∈ C − Supp(μ)
where Ind μ (z) = 0 and Int(μ), interior of μ, for the set of z where
Ind μ (z) = 0. The exterior of μ contains the non-compact connected component of C − Supp(μ), but may be strictly larger. These notions are not at
all related to those defined in Chap. III, no 1 with respect to an arbitrary
subset of C, but refer instead to what has been said at the end of Chap. III,
§ 4 (Jordan curve theorem) in the case of a “ simple ” curve, i.e. homeomorphic to the unit circle T. This supposedly simple case being already quite