§ 1. Integrals of Holomorphic Functions
19
σ : I × I −→ G
i.e. satisfying the following conditions :
(a) σ is of class C
2 on the open interior of I × I ⊂ R
2 ;
(b) the partial derivatives of order ≤ 2 of σ can be extended by continuity
16
to I × I.
Such a map defines two families of C
2 paths on G, namely
μ s : t −→ σ(s, t)
(3.1)
and
ν t : s −→ σ(s, t) .
(3.2)
As σ is continuous, the family of paths μ s may be regarded as a “ deformation ”
of μ 0 to μ 1 . The fact that a path can be deformed into another one in this
way, or even by a merely continuous function σ from I × I to G, is expressed
by saying that the two paths considered are homotopic. A first useful case is
that of a fixed-endpoint homotopy of μ 0 to μ 1 : then suppose that
μ s (0) = σ(s, 0) and μ s (1) = σ(s, 1)
are independent of s. Another case occurs when, μ 0 and μ 1 being closed, the
intermediate paths μ s stay closed during the deformation:
σ(0, t) = σ(1, t) for all t ;
μ 0 is said to be homotopic through closed paths to μ 1 .
Apart from these cases, the homotopy condition is always fulfilled (hence
uninteresting) because, on the one hand, every path μ is homotopic to a
“ constant ” path by σ(s, t) = μ [(1 − s)t], and because, on the other, two
“ constant ” paths are always homotopic as can be seen by connecting the
former to the latter by a path in G and by shifting the former along it to
take it onto the latter.
Despite being somewhat abstract, fixed-endpoint homotopy can be interpreted in an interesting way. Remark first that the set C
0 (I) of all continuous paths I −→ C,
17 equipped with the norm μI = sup |μ(t)| and the
obvious algebraic operations (addition, multiplication by a complex number),
is a complete normed vector space (Cauchy’s criterion for uniform convergence), i.e. a Banach space (Chap. III, Appendix, no 5). Paths can, therefore,
be defined in C
0 (I) as in any topological space : they are continuous maps
h : [0, 1] = I −→ C
0 (I). Hence, for any s ∈ I, h(s) = μs is a path in C, and
setting σ(s, t) = μs(t), we get a map σ from I × I to C; t → σ(s, t) = μs(t) is
clearly continuous for all s.
16 As I × I is compact, this means precisely that they are uniformly continuous on
the open set ]0, 1[×]0, 1[ : Chap. V, § 2, n
◦ 2, Corollary 2 of Theorem 8.
17 Up to vocabulary, a continuous “ path ” is just a complex valued function defined
and continuous on I.
19
σ : I × I −→ G
i.e. satisfying the following conditions :
(a) σ is of class C
2 on the open interior of I × I ⊂ R
2 ;
(b) the partial derivatives of order ≤ 2 of σ can be extended by continuity
16
to I × I.
Such a map defines two families of C
2 paths on G, namely
μ s : t −→ σ(s, t)
(3.1)
and
ν t : s −→ σ(s, t) .
(3.2)
As σ is continuous, the family of paths μ s may be regarded as a “ deformation ”
of μ 0 to μ 1 . The fact that a path can be deformed into another one in this
way, or even by a merely continuous function σ from I × I to G, is expressed
by saying that the two paths considered are homotopic. A first useful case is
that of a fixed-endpoint homotopy of μ 0 to μ 1 : then suppose that
μ s (0) = σ(s, 0) and μ s (1) = σ(s, 1)
are independent of s. Another case occurs when, μ 0 and μ 1 being closed, the
intermediate paths μ s stay closed during the deformation:
σ(0, t) = σ(1, t) for all t ;
μ 0 is said to be homotopic through closed paths to μ 1 .
Apart from these cases, the homotopy condition is always fulfilled (hence
uninteresting) because, on the one hand, every path μ is homotopic to a
“ constant ” path by σ(s, t) = μ [(1 − s)t], and because, on the other, two
“ constant ” paths are always homotopic as can be seen by connecting the
former to the latter by a path in G and by shifting the former along it to
take it onto the latter.
Despite being somewhat abstract, fixed-endpoint homotopy can be interpreted in an interesting way. Remark first that the set C
0 (I) of all continuous paths I −→ C,
17 equipped with the norm μI = sup |μ(t)| and the
obvious algebraic operations (addition, multiplication by a complex number),
is a complete normed vector space (Cauchy’s criterion for uniform convergence), i.e. a Banach space (Chap. III, Appendix, no 5). Paths can, therefore,
be defined in C
0 (I) as in any topological space : they are continuous maps
h : [0, 1] = I −→ C
0 (I). Hence, for any s ∈ I, h(s) = μs is a path in C, and
setting σ(s, t) = μs(t), we get a map σ from I × I to C; t → σ(s, t) = μs(t) is
clearly continuous for all s.
16 As I × I is compact, this means precisely that they are uniformly continuous on
the open set ]0, 1[×]0, 1[ : Chap. V, § 2, n
◦ 2, Corollary 2 of Theorem 8.
17 Up to vocabulary, a continuous “ path ” is just a complex valued function defined
and continuous on I.