10
VIII – Cauchy Theory
Nevertheless, we know – this is (1.11) – that if μ : I −→ G is a map from
an interval I ⊂ R to G, i.e. a path
6 in G, then
d
dt
F [μ(t)] = F
[μ(t)] μ
(t) = f [μ(t)] μ
(t)
(2.4)
at all points where the derivative μ
(t) exists. If μ is of class C
1 , if I = [u, v]
and if μ(u) = a, μ(v) = z, the simplest version of FT, therefore, shows that
F (z) − F (a) =
I
f [μ(t)] μ
(t)dt
(2.5)
since the right hand side of (4) is a continuous function of t; formula (2) is
obtained for μ(t) = tz. This result still holds if μ is of class C
1/2 : formula (4)
holds at all points where μ has a derivative, hence outside some countable
subset of I, and the function f [μ(t)] μ
(t) is regulated; the continuous function
F [μ(t)] is, therefore, a primitive for the latter, and so (5) follows. A path of
class C
1/2 will also be said to be admissible.
If we set ζ = μ(t), ` a la Leibniz, then dζ = μ
(t)dt, so that on the right
hand side of (5) we integrate the expression f (ζ)dζ; this leads to define the
integral of f along a path μ by
μ
f (ζ)dζ =
I
f [μ(t)] μ
(t)dt
(2.6)
in the same way as in Chap. V, eq. (5.16) for Cauchy’s formula for a circle.
The notation introduced on the left hand side of (6) could be justified by
observing that choosing a subdivision of I = [u, v] by the points u = t 0 <
t 1 < . . . < t n = v and setting ζ i = μ (t i ), integral (5) is approximately equal
to
f (ζ i ) μ
(t i ) (t i+1 − t i )
and hence, by the mean value formula, no less approximately, to
f (ζ i ) (ζ i+1 − ζ i ); whence notation (6). The reader will easily be able to
add the ε necessary to correct this simplified argument by subdividing I so
that the functions considered are piecewise constant up to ε; see point (iv)
further down.
Note that integral (6) does not depend exclusively on the “ curve ” μ(I)
defined by μ(t) as t varies in I; indeed, the latter does not change if the map
6 Everyone uses the letter γ to denote a path. I will use the letter μ because
(i) computer keyboard “ designers ” have had the good idea to include one and
only one Greek letter, namely μ, (ii), more seriously, as we will see a bit further
down, the function μ(t) occurs by way of the Radon (or Stieltjes) measure dμ(t) =
μ
(t)dt which it defines.
VIII – Cauchy Theory
Nevertheless, we know – this is (1.11) – that if μ : I −→ G is a map from
an interval I ⊂ R to G, i.e. a path
6 in G, then
d
dt
F [μ(t)] = F
[μ(t)] μ
(t) = f [μ(t)] μ
(t)
(2.4)
at all points where the derivative μ
(t) exists. If μ is of class C
1 , if I = [u, v]
and if μ(u) = a, μ(v) = z, the simplest version of FT, therefore, shows that
F (z) − F (a) =
I
f [μ(t)] μ
(t)dt
(2.5)
since the right hand side of (4) is a continuous function of t; formula (2) is
obtained for μ(t) = tz. This result still holds if μ is of class C
1/2 : formula (4)
holds at all points where μ has a derivative, hence outside some countable
subset of I, and the function f [μ(t)] μ
(t) is regulated; the continuous function
F [μ(t)] is, therefore, a primitive for the latter, and so (5) follows. A path of
class C
1/2 will also be said to be admissible.
If we set ζ = μ(t), ` a la Leibniz, then dζ = μ
(t)dt, so that on the right
hand side of (5) we integrate the expression f (ζ)dζ; this leads to define the
integral of f along a path μ by
μ
f (ζ)dζ =
I
f [μ(t)] μ
(t)dt
(2.6)
in the same way as in Chap. V, eq. (5.16) for Cauchy’s formula for a circle.
The notation introduced on the left hand side of (6) could be justified by
observing that choosing a subdivision of I = [u, v] by the points u = t 0 <
t 1 < . . . < t n = v and setting ζ i = μ (t i ), integral (5) is approximately equal
to
f (ζ i ) μ
(t i ) (t i+1 − t i )
and hence, by the mean value formula, no less approximately, to
f (ζ i ) (ζ i+1 − ζ i ); whence notation (6). The reader will easily be able to
add the ε necessary to correct this simplified argument by subdividing I so
that the functions considered are piecewise constant up to ε; see point (iv)
further down.
Note that integral (6) does not depend exclusively on the “ curve ” μ(I)
defined by μ(t) as t varies in I; indeed, the latter does not change if the map
6 Everyone uses the letter γ to denote a path. I will use the letter μ because
(i) computer keyboard “ designers ” have had the good idea to include one and
only one Greek letter, namely μ, (ii), more seriously, as we will see a bit further
down, the function μ(t) occurs by way of the Radon (or Stieltjes) measure dμ(t) =
μ
(t)dt which it defines.