§ 2. Cauchy’s Integral Formulas
53
ψ
[ν(t)] ν
(t) =
d
dt
ψ [ν(t)] = μ
(t) ;
so finally
ν
=
f [μ(t)] μ
(t)dt =
μ
ω
(5.22)
as expected.
This result is independent from residue theory, but suppose that μ is a
closed path in U − S null-homotopic in U . In this case, ν is a closed path in
V − ϕ(S), clearly null-homotopic in V . Then the residue formula (Theorem
5) shows that
Ind ν (b) Res(, b) =
Ind μ (a) Res(ω, a)
follows from (22). This result can be applied to the case S = {a}, ϕ(a) = b,
where a is any point in U , and where f is a holomorphic function on U − {a},
for example 1/(z − a). The residues being equal, we conclude that
Ind ν [ϕ(a)] = Ind μ (a) .
(5.23)
This may seem obvious geometrically but is not so, especially as ϕ could a
priori transform a counterclockwise path μ around a into a counterclockwise
path ν around b = ϕ(a).
Hence a conformal representation ϕ leaves the “ rotation direction around
a point ” of a closed path invariant; as will be seen in the next chapter, in a far
more general situation, this is due to the fact that the Jacobian of ϕ = p + iq,
regarded as a map R
2 to R
2 , namely
J ϕ (z) = D 1 p.D 2 q − D 2 p.D 1 q = |ϕ
(z)|
2 ,
is positive.
(vi) Functions on the Riemann sphere. A new set
ˆ
C = C ∪ {∞} ,
obtained by adding to C an element written ∞, whose choice and nature
matter little, was introduced above – reread Hardy in Chapter II, end of n
◦ 2.
Having done this, define a topology on ˆ
C by setting U ⊂ ˆ
C to be open if U ∩ C
is open in C in the usual sense and if the exterior of a disc is contained in U
when ∞ ∈ U . Open subsets containing the point ∞ are, therefore, precisely
the complements of the compact subsets of C in ˆ
C. Axioms about unions and
intersections of open sets are immediately seen to be satisfied. This topology
of ˆ
C allows us to define the notions of limit and continuity; for example,
saying that a sequence of points z n ∈ C approaches ∞ with respect to the
topology of ˆ
C means that |z n | increases indefinitely since we need to intimate
53
ψ
[ν(t)] ν
(t) =
d
dt
ψ [ν(t)] = μ
(t) ;
so finally
ν
=
f [μ(t)] μ
(t)dt =
μ
ω
(5.22)
as expected.
This result is independent from residue theory, but suppose that μ is a
closed path in U − S null-homotopic in U . In this case, ν is a closed path in
V − ϕ(S), clearly null-homotopic in V . Then the residue formula (Theorem
5) shows that
Ind ν (b) Res(, b) =
Ind μ (a) Res(ω, a)
follows from (22). This result can be applied to the case S = {a}, ϕ(a) = b,
where a is any point in U , and where f is a holomorphic function on U − {a},
for example 1/(z − a). The residues being equal, we conclude that
Ind ν [ϕ(a)] = Ind μ (a) .
(5.23)
This may seem obvious geometrically but is not so, especially as ϕ could a
priori transform a counterclockwise path μ around a into a counterclockwise
path ν around b = ϕ(a).
Hence a conformal representation ϕ leaves the “ rotation direction around
a point ” of a closed path invariant; as will be seen in the next chapter, in a far
more general situation, this is due to the fact that the Jacobian of ϕ = p + iq,
regarded as a map R
2 to R
2 , namely
J ϕ (z) = D 1 p.D 2 q − D 2 p.D 1 q = |ϕ
(z)|
2 ,
is positive.
(vi) Functions on the Riemann sphere. A new set
ˆ
C = C ∪ {∞} ,
obtained by adding to C an element written ∞, whose choice and nature
matter little, was introduced above – reread Hardy in Chapter II, end of n
◦ 2.
Having done this, define a topology on ˆ
C by setting U ⊂ ˆ
C to be open if U ∩ C
is open in C in the usual sense and if the exterior of a disc is contained in U
when ∞ ∈ U . Open subsets containing the point ∞ are, therefore, precisely
the complements of the compact subsets of C in ˆ
C. Axioms about unions and
intersections of open sets are immediately seen to be satisfied. This topology
of ˆ
C allows us to define the notions of limit and continuity; for example,
saying that a sequence of points z n ∈ C approaches ∞ with respect to the
topology of ˆ
C means that |z n | increases indefinitely since we need to intimate