§ 1. Integrals of Holomorphic Functions
27
immediately transferred to holomorphic or harmonic functions on V . As relation g [f (z)] = z shows that the derivatives of f and g are mutually inverse
at corresponding points, it follows that f
(z) = 0 for all z ∈ G. Conversely, if
this condition is satisfied, then f , though not necessarily a global homeomorphism (for this f would need to be injective) transforms every open subset
of U , and in particular U itself, into an open subset of C. This was proved
in Chapter III, § 5, n
◦ 24 using the local inversion theorem : set f = p + iq
to be the map taking the point (x, y) ∈ R
2 to the point (ξ, η) such that
ξ + iη = f (x + iy) can also be written
ξ = p(x, y) , η = q(x, y) ,
so that, by Cauchy’s equations, its Jacobian
J f (x, y) = D 1 p(x, y)D 2 q(x, y) − D 2 p(x, y)D 1 q(x, y)
is equal to
D 1 p(x, y)
2 + D 1 q(x, y)
2 = |f
(z)|
2 ,
and hence is non-zero. This proves the result. Moreover, if f : U −→ V =
f (U ) is assumed to be injective, then f is a homeomorphism [since the inverse
image of an open set U
⊂ U under f
−1 is then f (U
), and so is open] and
the local inversion theorem shows that, like f , the inverse map g : V −→ U
is of class C
1 as a function of two real variables. It is holomorphic since the
relation g [f (z)] = z implies that the Jacobian matrix of g at ζ = f (z) is the
inverse of that of f at z. Now, holomorphic functions are characterized by
the fact that, at all points, their Jacobian matrix is of the form
a b
−b a
.
It is, therefore, sufficient to check that the inverse of such a matrix is also of
the same type. More simply: the inverse of any C-linear map is C-linear. All
this has been proved in Chapter III, § 5, but it is worth recalling here. We will
see in n
◦ 5 (Theorem 7) that the relation f
(z) = 0 is in fact a consequence
of the injectivity of f , in other words that conformal representations are just
bijective holomorphic maps.
The fact that a simply connected domain G other than C is isomorphic
to the unit disc is one of the most famous results of Riemann; while being
based on a method going far beyond the framework of holomorphic functions
(PDE specialists’ “ Dirichlet’s principle ”), his proof was not really satisfactory. Simpler proofs have since then been found
20 and the behaviour of a
conformal representation f of G on the unit disc in the neighbourhood of the
boundary of G has been widely studied; if, for example, G is bounded and
20 See for instance Chap. 14 in Rudin and for examples Chap. X in Dieudonn´ e,
Analyse infinit´ esimale.
27
immediately transferred to holomorphic or harmonic functions on V . As relation g [f (z)] = z shows that the derivatives of f and g are mutually inverse
at corresponding points, it follows that f
(z) = 0 for all z ∈ G. Conversely, if
this condition is satisfied, then f , though not necessarily a global homeomorphism (for this f would need to be injective) transforms every open subset
of U , and in particular U itself, into an open subset of C. This was proved
in Chapter III, § 5, n
◦ 24 using the local inversion theorem : set f = p + iq
to be the map taking the point (x, y) ∈ R
2 to the point (ξ, η) such that
ξ + iη = f (x + iy) can also be written
ξ = p(x, y) , η = q(x, y) ,
so that, by Cauchy’s equations, its Jacobian
J f (x, y) = D 1 p(x, y)D 2 q(x, y) − D 2 p(x, y)D 1 q(x, y)
is equal to
D 1 p(x, y)
2 + D 1 q(x, y)
2 = |f
(z)|
2 ,
and hence is non-zero. This proves the result. Moreover, if f : U −→ V =
f (U ) is assumed to be injective, then f is a homeomorphism [since the inverse
image of an open set U
⊂ U under f
−1 is then f (U
), and so is open] and
the local inversion theorem shows that, like f , the inverse map g : V −→ U
is of class C
1 as a function of two real variables. It is holomorphic since the
relation g [f (z)] = z implies that the Jacobian matrix of g at ζ = f (z) is the
inverse of that of f at z. Now, holomorphic functions are characterized by
the fact that, at all points, their Jacobian matrix is of the form
a b
−b a
.
It is, therefore, sufficient to check that the inverse of such a matrix is also of
the same type. More simply: the inverse of any C-linear map is C-linear. All
this has been proved in Chapter III, § 5, but it is worth recalling here. We will
see in n
◦ 5 (Theorem 7) that the relation f
(z) = 0 is in fact a consequence
of the injectivity of f , in other words that conformal representations are just
bijective holomorphic maps.
The fact that a simply connected domain G other than C is isomorphic
to the unit disc is one of the most famous results of Riemann; while being
based on a method going far beyond the framework of holomorphic functions
(PDE specialists’ “ Dirichlet’s principle ”), his proof was not really satisfactory. Simpler proofs have since then been found
20 and the behaviour of a
conformal representation f of G on the unit disc in the neighbourhood of the
boundary of G has been widely studied; if, for example, G is bounded and
20 See for instance Chap. 14 in Rudin and for examples Chap. X in Dieudonn´ e,
Analyse infinit´ esimale.