22
VIII – Cauchy Theory
All this obviously supposes that only values of s such that μ+sν ∈ C
1/2 (I; G)
are used. The reader will probably be under the impression of having come
across similar calculations above. A more or less vague form of this idea is
due to Cauchy : in his work on the series expansion of f [μ(t) + sν(t)] with
respect to s, he calculated the coefficient of s. His calculations have long since
disappeared from textbooks. This is a good reason for reintroducing them by
rectifying his simplistic, but ultimately correct idea, since it reappears in a
very generalized form in the version of the calculus of variations for example
found in H. Cartan, Differential Calculus, where a function of the following
form is differentiated with respect to μ:
F (μ) =
b
a
f [t, μ(t), μ
(t)] dt .
On the other hand, note that in the presence of a formula such as (5), the
idea of deducing an expression for F (μ+sν) by applying the FT immediately
arises. This will be done a bit later.
Exercise 1. Let μ 0 and μ 1 be two paths on G and σ : I × I −→ G a
homotopy from μ 0 to μ 1 ; write F (s) for the integral of the function f (z)
along the path μ s . assuming σ to be of class C
2 , find a formula similar to (5)
for F
(s).
(iii) Effects of a linear homotopy on an integral. We can now return to
the behaviour of an integral when an integration path μ 0 is deformed into
a path μ 1 without leaving the domain G where the function f to be integrated is defined and holomorphic. The difficulty is that, for 0 < s < 1, the
intermediary paths μ s are continuous, but not necessarily admissible. We can
get around it by altering the homotopy so that the μ s become admissible or,
equivalently, by showing that it is possible to go from μ 0 to μ 1 by a succession
of linear homotopies between admissible paths, i.e. of the form
σ(s, t) = (1 − s)μ 0 (t) + sμ 1 (t) = μ s (t) ,
(3.6)
where s, t ∈ I = [0, 1]. There is always such a homotopy when μ 1 is sufficiently
near μ 0 in the sense of uniform convergence : if R > 0 is the distance from
μ 0 (I) to the border of G, then σ(s, t) ∈ G for all s, t ∈ I provided μ 1 −μ 0 I <
R.
However, path (6) is of the form μ 0 + sν with
ν(t) = μ 1 (t) − μ 0 (t).
It is, therefore, possible to apply (5) to the function F (μ s ), whence
d
ds
μs
f (ζ)dζ = f [μ s (t)] [μ 1 (t) − μ 0 (t)]
t=1
t=0
.
(3.7)
Suppose first that it is a fixed-endpoint homotopy. Then, μ 1 (t) − μ 0 (t) = 0
for t = 0 or 1, the derivative is zero for all s and the integral is, therefore,
independent of s ∈ [0, 1]. In other words, in this case
VIII – Cauchy Theory
All this obviously supposes that only values of s such that μ+sν ∈ C
1/2 (I; G)
are used. The reader will probably be under the impression of having come
across similar calculations above. A more or less vague form of this idea is
due to Cauchy : in his work on the series expansion of f [μ(t) + sν(t)] with
respect to s, he calculated the coefficient of s. His calculations have long since
disappeared from textbooks. This is a good reason for reintroducing them by
rectifying his simplistic, but ultimately correct idea, since it reappears in a
very generalized form in the version of the calculus of variations for example
found in H. Cartan, Differential Calculus, where a function of the following
form is differentiated with respect to μ:
F (μ) =
b
a
f [t, μ(t), μ
(t)] dt .
On the other hand, note that in the presence of a formula such as (5), the
idea of deducing an expression for F (μ+sν) by applying the FT immediately
arises. This will be done a bit later.
Exercise 1. Let μ 0 and μ 1 be two paths on G and σ : I × I −→ G a
homotopy from μ 0 to μ 1 ; write F (s) for the integral of the function f (z)
along the path μ s . assuming σ to be of class C
2 , find a formula similar to (5)
for F
(s).
(iii) Effects of a linear homotopy on an integral. We can now return to
the behaviour of an integral when an integration path μ 0 is deformed into
a path μ 1 without leaving the domain G where the function f to be integrated is defined and holomorphic. The difficulty is that, for 0 < s < 1, the
intermediary paths μ s are continuous, but not necessarily admissible. We can
get around it by altering the homotopy so that the μ s become admissible or,
equivalently, by showing that it is possible to go from μ 0 to μ 1 by a succession
of linear homotopies between admissible paths, i.e. of the form
σ(s, t) = (1 − s)μ 0 (t) + sμ 1 (t) = μ s (t) ,
(3.6)
where s, t ∈ I = [0, 1]. There is always such a homotopy when μ 1 is sufficiently
near μ 0 in the sense of uniform convergence : if R > 0 is the distance from
μ 0 (I) to the border of G, then σ(s, t) ∈ G for all s, t ∈ I provided μ 1 −μ 0 I <
R.
However, path (6) is of the form μ 0 + sν with
ν(t) = μ 1 (t) − μ 0 (t).
It is, therefore, possible to apply (5) to the function F (μ s ), whence
d
ds
μs
f (ζ)dζ = f [μ s (t)] [μ 1 (t) − μ 0 (t)]
t=1
t=0
.
(3.7)
Suppose first that it is a fixed-endpoint homotopy. Then, μ 1 (t) − μ 0 (t) = 0
for t = 0 or 1, the derivative is zero for all s and the integral is, therefore,
independent of s ∈ [0, 1]. In other words, in this case