14
VIII – Cauchy Theory
(iv) A necessary and sufficient condition for a primitive. Let us return
to a holomorphic function f on a domain G of C. If it admits a primitive F
on G if a point a of G is chosen, then, as seen at the start of this n
◦ ,
F (z) − F (a) =
μ
f (ζ)dζ
(2.12)
for any admissible path μ connecting a to z in G. Hence, the integral of f
along such a path depends only on its endpoints.
In the general case, it may be tempting to construct a primitive by applying the previous formula and choosing its value at a arbitrarily, for example
F (a) = 0. But this definition of F is totally ambiguous : the value of the
integral can very well depend on the choice of the path μ connecting a to
z in G as the case
9 of the function 1/z already shows. Hence, the notation
F (z) used is, a priori, not well-defined; the only reasonable notation is to set
F (μ) =
μ
f (ζ)dζ
(2.13)
for every integration path
10 μ. As in the case of the logarithm of a complex
number z = 0 (Chap. IV, § 4 or Chap. VII, n
◦ 16), we get a set F(z) of
possible values of the function sought, namely all the numbers obtained by
integrating f along a path μ connecting a to z in G or, in the case of the
logarithm, all the numbers obtained by using a uniform branch of Log along
a path connecting a fixed point a to the point z (as we will see, this amounts
to integrating 1/ζ along this path). But as in the case of the logarithm, the
problem is to construct a truly holomorphic function F , and in particular
continuous, such that F (z) ∈ F(z) for all z ∈ G. In the case of the logarithm,
we have seen this to to be possible if and only if the following condition is
satisfied : if, for some given a ∈ G and a varying path μ : I −→ G with initial
9 If it was independent of the path in this case, the integral of 1/ζ along the path
t → exp(2πit) connecting the point a = 1 to the point z = 1 would be equal
to that obtained by integrating along the “ constant ” path t → 1, i.e. to 0.
However, the integral over the circle is obtained by integrating the function 2πi
over [0, 1] and hence is not zero. Integrals in 1/z will be discussed in detail later.
10 Mathematicians who invented the “ calculus of variations ” almost three centuries
ago already had the idea of considering functions of a curve varying in the plane,
on a surface or in space, a curve along which a given function is integrated; a
century ago, the mathematician Vito Volterra used to call these line functions.
For a “ smooth ” surface S in R
3 , we can, for example, try to find the curves
of minimal length drawn on S connecting two given points : the geodesics; the
length of a curve μ is given by (4.8) and by comparing it to a curve “ infinitely
near ” to it we get a differential equation characterizing the geodesics. Quite an
old problem in mechanics consists in find a curve connecting A to B for two
given points A and B such that the time taken to go from A to B by an object
moving along it under the action of gravity is minimum. Fermat already knew
that the trajectory of a light beam going from point A to a point B through a
medium whose retractive index varies is the one that minimizes travel time. Etc.
VIII – Cauchy Theory
(iv) A necessary and sufficient condition for a primitive. Let us return
to a holomorphic function f on a domain G of C. If it admits a primitive F
on G if a point a of G is chosen, then, as seen at the start of this n
◦ ,
F (z) − F (a) =
μ
f (ζ)dζ
(2.12)
for any admissible path μ connecting a to z in G. Hence, the integral of f
along such a path depends only on its endpoints.
In the general case, it may be tempting to construct a primitive by applying the previous formula and choosing its value at a arbitrarily, for example
F (a) = 0. But this definition of F is totally ambiguous : the value of the
integral can very well depend on the choice of the path μ connecting a to
z in G as the case
9 of the function 1/z already shows. Hence, the notation
F (z) used is, a priori, not well-defined; the only reasonable notation is to set
F (μ) =
μ
f (ζ)dζ
(2.13)
for every integration path
10 μ. As in the case of the logarithm of a complex
number z = 0 (Chap. IV, § 4 or Chap. VII, n
◦ 16), we get a set F(z) of
possible values of the function sought, namely all the numbers obtained by
integrating f along a path μ connecting a to z in G or, in the case of the
logarithm, all the numbers obtained by using a uniform branch of Log along
a path connecting a fixed point a to the point z (as we will see, this amounts
to integrating 1/ζ along this path). But as in the case of the logarithm, the
problem is to construct a truly holomorphic function F , and in particular
continuous, such that F (z) ∈ F(z) for all z ∈ G. In the case of the logarithm,
we have seen this to to be possible if and only if the following condition is
satisfied : if, for some given a ∈ G and a varying path μ : I −→ G with initial
9 If it was independent of the path in this case, the integral of 1/ζ along the path
t → exp(2πit) connecting the point a = 1 to the point z = 1 would be equal
to that obtained by integrating along the “ constant ” path t → 1, i.e. to 0.
However, the integral over the circle is obtained by integrating the function 2πi
over [0, 1] and hence is not zero. Integrals in 1/z will be discussed in detail later.
10 Mathematicians who invented the “ calculus of variations ” almost three centuries
ago already had the idea of considering functions of a curve varying in the plane,
on a surface or in space, a curve along which a given function is integrated; a
century ago, the mathematician Vito Volterra used to call these line functions.
For a “ smooth ” surface S in R
3 , we can, for example, try to find the curves
of minimal length drawn on S connecting two given points : the geodesics; the
length of a curve μ is given by (4.8) and by comparing it to a curve “ infinitely
near ” to it we get a differential equation characterizing the geodesics. Quite an
old problem in mechanics consists in find a curve connecting A to B for two
given points A and B such that the time taken to go from A to B by an object
moving along it under the action of gravity is minimum. Fermat already knew
that the trajectory of a light beam going from point A to a point B through a
medium whose retractive index varies is the one that minimizes travel time. Etc.