18
VIII – Cauchy Theory
satisfying the following conditions :
(i) H(z, 0) = a, H(z, 1) = z for all z ∈ G,
(ii) H is of class C
2
in the sense specified in the previous footnote. It is the case of the map
(z, t) → tz for a star domain about the origin.
The existence of a continuous, but not necessarily C
2 , map H from G × I
to G satisfying (i) for some point a ∈ G is expressed by saying that the
domain G is contractible onto a. Setting H t (z) = H(1 − t, z), we then get a
one-parameter family of continuous maps H t indexed by t from G into itself
starting with the identity map z → z and, at the end of the process, taking G
onto the point a; under the “ contraction ”, each z ∈ G describes a trajectory
t → H(1 − t, z) which takes it from its initial position to the point a. It can
be shown that if there is a contraction of class C
0 from G onto a point, then
there also is one that is C
2 and even C
∞ ; while not being easy, it is not
very difficult to prove. Hence, if we admit this point,
14 we get a more general
result than that of n
◦ 1 regarding star domains, but it will in its turn be
generalized (?) further down :
Theorem 2. Every holomorphic function defined on a contractible domain
G ⊂ C has a primitive on G.
Corollary. An annulus r < |z| < R is not contractible (which is physically
obvious), since the function 1/z does not have primitives : its integral along
the circle centered at 0 is obtained by integrating 2πi over [0, 1], and so is
equal to 2πi despite the closure of the integration path. However, C − R − is
contractible and even a star domain (consider the homotheties with centre 1),
which explain why the function 1/z has a primitive on the domain, namely
any uniform branch of the pseudo-function Log z.
3 – Homotopy Invariance of Integrals
(i) Homotopic paths. Computation (2.15) to differentiate under
sign would
continue to hold if the function H(x, y, t) was replaced by a function of multiple real variables with values in G. The simplest case is that of a C
2 map,
15
14 It is in fact unnecessary as theorem 2 is a consequence of theorem 3 which will
be proved later. The relevance of theorem 2 as stated here only lie in its proof
and, as such, is only a calculus exercise.
15 For reasons stated in chapter 9 – similarities between curvilinear integrals (dimension 1) and surface integrals (dimension 2) –, σ may be called a 2 dimensional
path in C ; the reader will easily generalize to all dimensions. There is no orthodox terminology; some, like Serge Lang, talk wrongly of a 2 dimensional simplex
as in algebraic topology. Ours suggests that such a “ path ” takes us in a continuous manner from the usual path μ0 : t → σ(0, t) to another one, μ1 : t → σ(1, t),
just like an usual one-dimensional path takes us in a continuous manner from a
point, a 0-dimensional path, to another one.
VIII – Cauchy Theory
satisfying the following conditions :
(i) H(z, 0) = a, H(z, 1) = z for all z ∈ G,
(ii) H is of class C
2
in the sense specified in the previous footnote. It is the case of the map
(z, t) → tz for a star domain about the origin.
The existence of a continuous, but not necessarily C
2 , map H from G × I
to G satisfying (i) for some point a ∈ G is expressed by saying that the
domain G is contractible onto a. Setting H t (z) = H(1 − t, z), we then get a
one-parameter family of continuous maps H t indexed by t from G into itself
starting with the identity map z → z and, at the end of the process, taking G
onto the point a; under the “ contraction ”, each z ∈ G describes a trajectory
t → H(1 − t, z) which takes it from its initial position to the point a. It can
be shown that if there is a contraction of class C
0 from G onto a point, then
there also is one that is C
2 and even C
∞ ; while not being easy, it is not
very difficult to prove. Hence, if we admit this point,
14 we get a more general
result than that of n
◦ 1 regarding star domains, but it will in its turn be
generalized (?) further down :
Theorem 2. Every holomorphic function defined on a contractible domain
G ⊂ C has a primitive on G.
Corollary. An annulus r < |z| < R is not contractible (which is physically
obvious), since the function 1/z does not have primitives : its integral along
the circle centered at 0 is obtained by integrating 2πi over [0, 1], and so is
equal to 2πi despite the closure of the integration path. However, C − R − is
contractible and even a star domain (consider the homotheties with centre 1),
which explain why the function 1/z has a primitive on the domain, namely
any uniform branch of the pseudo-function Log z.
3 – Homotopy Invariance of Integrals
(i) Homotopic paths. Computation (2.15) to differentiate under
sign would
continue to hold if the function H(x, y, t) was replaced by a function of multiple real variables with values in G. The simplest case is that of a C
2 map,
15
14 It is in fact unnecessary as theorem 2 is a consequence of theorem 3 which will
be proved later. The relevance of theorem 2 as stated here only lie in its proof
and, as such, is only a calculus exercise.
15 For reasons stated in chapter 9 – similarities between curvilinear integrals (dimension 1) and surface integrals (dimension 2) –, σ may be called a 2 dimensional
path in C ; the reader will easily generalize to all dimensions. There is no orthodox terminology; some, like Serge Lang, talk wrongly of a 2 dimensional simplex
as in algebraic topology. Ours suggests that such a “ path ” takes us in a continuous manner from the usual path μ0 : t → σ(0, t) to another one, μ1 : t → σ(1, t),
just like an usual one-dimensional path takes us in a continuous manner from a
point, a 0-dimensional path, to another one.