26
VIII – Cauchy Theory
function g on G such that e
g(z) = f (z) for all z ∈ G; it is unique up to
addition of a multiple of 2πi.
Since f does not vanish on G, the function f
/f is defined and holomorphic
on G, and hence admits a primitive g; then (e
g )
= g
e
g = e
g f
/f , i.e. (e
g )
f −
e
g f
= 0, hence (e
g /f )
= 0, so that the function e
g is proportional to f .
Adding a constant to g, e
g = f may be assumed. Any other holomorphic
solution g 1 must satisfy the condition g 1 (z) − g(z) ∈ 2πiZ for all z, which
obviously requires the left hand side to be a constant, qed.
The relation e
g = f means that g(z) ∈ Log f (z) holds for all z ∈ G, where
Log w denotes the set z ∈ C such that exp(z) = w (Chapter IV, § 4). Such a
function g is called a uniform branch of the pseudo-function Log f (z). Then
g(z) = log |f (z)| + i. Arg f (z) ,
where, at all points, the argument of f (z) must be chosen so that it is a
continuous function of z.
The definition of uniform branches of the no less pseudo-functions f (z)
s ,
where s ∈ C is given and is not an integer, follows from such branches g : these
are the functions e
s.g(z) . For s = 1/p with p integer, we thus get holomorphic
solutions of the equation h(z)
p = f (z); they can be deduced from any one of
them by taking its product with a p
th root of unity.
All this supposes that G is simply connected. The case of the function
f (z) = 1/z on G = C − {0} shows that this assumption is essential. In fact,
Corollary 1 can be shown to characterize simply connected domains, but this
result is rarely used.
Corollary 3. Let f be a holomorphic functions on a domain G. The integral
of f along any null-homotopic closed path, i.e. homotopic to a point in G, is
zero.
This result explains Theorem 2 whose proof used a homotopy of class C
2 ,
an assumption now unnecessary.
Any contractible domain G is simply connected. Indeed, if σ is a contraction to a point a ∈ G and μ a closed path in G, then μ is homotopic through
closed paths to the constant path t → a under the map (s, t) → σ [1 − s, μ(t)].
This trivial result has a far less trivial converse : any simply connected
domain G in C is not only contractible, but also homeomorphic to the unit
disc |z| < 1; except when G = C, a case excluded by Liouville’s theorem
on integral functions, even in G, there is a holomorphic function mapping G
bijectively onto the open disc D : |z| < 1 and whose inverse is holomorphic
(Riemann). Any holomorphic bijection f : U −→ V from an open set onto
another whose inverse g : V −→ U is holomorphic is called a conformal representation of U on V . The existence of such a representation means that U
and V are “ isomorphic ” from the point of view of analytic function theory :
everything that holds for holomorphic or harmonic functions on U can be
VIII – Cauchy Theory
function g on G such that e
g(z) = f (z) for all z ∈ G; it is unique up to
addition of a multiple of 2πi.
Since f does not vanish on G, the function f
/f is defined and holomorphic
on G, and hence admits a primitive g; then (e
g )
= g
e
g = e
g f
/f , i.e. (e
g )
f −
e
g f
= 0, hence (e
g /f )
= 0, so that the function e
g is proportional to f .
Adding a constant to g, e
g = f may be assumed. Any other holomorphic
solution g 1 must satisfy the condition g 1 (z) − g(z) ∈ 2πiZ for all z, which
obviously requires the left hand side to be a constant, qed.
The relation e
g = f means that g(z) ∈ Log f (z) holds for all z ∈ G, where
Log w denotes the set z ∈ C such that exp(z) = w (Chapter IV, § 4). Such a
function g is called a uniform branch of the pseudo-function Log f (z). Then
g(z) = log |f (z)| + i. Arg f (z) ,
where, at all points, the argument of f (z) must be chosen so that it is a
continuous function of z.
The definition of uniform branches of the no less pseudo-functions f (z)
s ,
where s ∈ C is given and is not an integer, follows from such branches g : these
are the functions e
s.g(z) . For s = 1/p with p integer, we thus get holomorphic
solutions of the equation h(z)
p = f (z); they can be deduced from any one of
them by taking its product with a p
th root of unity.
All this supposes that G is simply connected. The case of the function
f (z) = 1/z on G = C − {0} shows that this assumption is essential. In fact,
Corollary 1 can be shown to characterize simply connected domains, but this
result is rarely used.
Corollary 3. Let f be a holomorphic functions on a domain G. The integral
of f along any null-homotopic closed path, i.e. homotopic to a point in G, is
zero.
This result explains Theorem 2 whose proof used a homotopy of class C
2 ,
an assumption now unnecessary.
Any contractible domain G is simply connected. Indeed, if σ is a contraction to a point a ∈ G and μ a closed path in G, then μ is homotopic through
closed paths to the constant path t → a under the map (s, t) → σ [1 − s, μ(t)].
This trivial result has a far less trivial converse : any simply connected
domain G in C is not only contractible, but also homeomorphic to the unit
disc |z| < 1; except when G = C, a case excluded by Liouville’s theorem
on integral functions, even in G, there is a holomorphic function mapping G
bijectively onto the open disc D : |z| < 1 and whose inverse is holomorphic
(Riemann). Any holomorphic bijection f : U −→ V from an open set onto
another whose inverse g : V −→ U is holomorphic is called a conformal representation of U on V . The existence of such a representation means that U
and V are “ isomorphic ” from the point of view of analytic function theory :
everything that holds for holomorphic or harmonic functions on U can be