§ 1. Integrals of Holomorphic Functions
11
μ is replaced by ν(t) = μ (ϕ(t)), where ϕ is a surjective map from an interval
J to I; supposing, for simplicity’s sake, that ϕ is of class C
1 ,
ν
f (z)dz =
J
f [μ (ϕ(t))] μ
[ϕ(t)] ϕ
(t)dt
by the chain rule; hence, setting F (t) = f [μ(t)] μ
(t),
μ
f (z)dz =
I
F (t)dt ,
ν
f (z)dz =
J
F [ϕ
(t)] ϕ
(t)dt ,
As I = ϕ(J), the equality of these two integrals seems to follow from the
change of variables formula for integrals (Chapter V, § 6, n
◦ 19). But the
latter concerns oriented integrals. Hence, the equality
μ
f (z)dz =
ν
f (z)dz
supposes that ϕ maps the initial (resp. terminal) point of J onto the initial
(resp. terminal) point of I. Otherwise, the previous relation only holds up
to sign. In practice, only strictly increasing “ changes of parameter ” ϕ are
considered. This avoids difficulties and reduces the question to paths for which
I = [0, 1]. From now on, we will suppose this to be the case, unless stated
otherwise.
Admissible paths or those of class C
1/2 cover all cases that might arise. In
practice, I can almost always be divided into intervals on which the function
μ is C
1 , if not linear. But, once the notion of a primitive for a regulated
function has been understood, using these “ piecewise ” paths of class C
1
(or linear), as they are called, is not any easier than using paths of class
C
1/2 . Applying Theorem 12 bis of Chap. V, n
◦ 13, we see that if a path is
considered to be the trajectory of a moving object, then admissible paths can
be characterized by imposing the following conditions upon them:
(a) the map μ is continuous,
(b) it has a right and left derivative at every point t,
(c) these are equal outside some countable subset D of I (for example, it
may be that D = I ∩ Q, but it is better to avoid this kind of paths in
practical calculations. . . ),
(d) The right (or left) derivative is a regulated function of t, i.e. has righthand and left-hand limits for all t or, equivalently, is the uniform limit
on I of step functions.
The trajectory defined by μ(t), possibly passing over the same point several times, therefore, admits a “ velocity vector ” μ
(t) outside D as it can
change direction (“ angular points ”) at points of D. It has a tangent at each
point where μ
(t) exists and is non-zero. As any driver attempting to park
between two cars knows, the case μ
(t) = 0 can entail a cusp point.
7
7 Example : t → (t
2 , t
3 ) at t = 0, with I = [−1, 1].
11
μ is replaced by ν(t) = μ (ϕ(t)), where ϕ is a surjective map from an interval
J to I; supposing, for simplicity’s sake, that ϕ is of class C
1 ,
ν
f (z)dz =
J
f [μ (ϕ(t))] μ
[ϕ(t)] ϕ
(t)dt
by the chain rule; hence, setting F (t) = f [μ(t)] μ
(t),
μ
f (z)dz =
I
F (t)dt ,
ν
f (z)dz =
J
F [ϕ
(t)] ϕ
(t)dt ,
As I = ϕ(J), the equality of these two integrals seems to follow from the
change of variables formula for integrals (Chapter V, § 6, n
◦ 19). But the
latter concerns oriented integrals. Hence, the equality
μ
f (z)dz =
ν
f (z)dz
supposes that ϕ maps the initial (resp. terminal) point of J onto the initial
(resp. terminal) point of I. Otherwise, the previous relation only holds up
to sign. In practice, only strictly increasing “ changes of parameter ” ϕ are
considered. This avoids difficulties and reduces the question to paths for which
I = [0, 1]. From now on, we will suppose this to be the case, unless stated
otherwise.
Admissible paths or those of class C
1/2 cover all cases that might arise. In
practice, I can almost always be divided into intervals on which the function
μ is C
1 , if not linear. But, once the notion of a primitive for a regulated
function has been understood, using these “ piecewise ” paths of class C
1
(or linear), as they are called, is not any easier than using paths of class
C
1/2 . Applying Theorem 12 bis of Chap. V, n
◦ 13, we see that if a path is
considered to be the trajectory of a moving object, then admissible paths can
be characterized by imposing the following conditions upon them:
(a) the map μ is continuous,
(b) it has a right and left derivative at every point t,
(c) these are equal outside some countable subset D of I (for example, it
may be that D = I ∩ Q, but it is better to avoid this kind of paths in
practical calculations. . . ),
(d) The right (or left) derivative is a regulated function of t, i.e. has righthand and left-hand limits for all t or, equivalently, is the uniform limit
on I of step functions.
The trajectory defined by μ(t), possibly passing over the same point several times, therefore, admits a “ velocity vector ” μ
(t) outside D as it can
change direction (“ angular points ”) at points of D. It has a tangent at each
point where μ
(t) exists and is non-zero. As any driver attempting to park
between two cars knows, the case μ
(t) = 0 can entail a cusp point.
7
7 Example : t → (t
2 , t
3 ) at t = 0, with I = [−1, 1].