§ 1. Integrals of Holomorphic Functions
3
§ 1. Integrals of Holomorphic Functions
1 – Preliminary Results
(i) The fundamental theorem (FT) of differential and integral calculus (Chap. V,
§ 3). In its simplest form, take a continuous function f on an interval I ⊂ R
and, choosing an a ∈ I, set
F (x) =
x
a
f (t)dt
a differentiable function such that F
(x) = f (x) for all x ∈ I is thus obtained.
Conversely, any primitive for f is given up to an additive constant by this
formula.
If we start with a regulated function
2 f – a less simple case –, the previous
formula defines a continuous function F admitting right-hand and left-hand
derivatives at every x ∈ I, given by
F
d (x) = lim
h = 0
h > 0
F (x + h) − F (x)
h
= f (x + 0) = lim
h = 0
h > 0
f (x + h)
and by a similar left-hand formula. In particular, the derivative F
(x) exists
outside some countable set D of discontinuous points of f . Conversely, if
there is regulated function f and a continuous function F in I which, outside
some countable subset of I, admits a derivative equal to f (x), then, up to
a constant, F is again given by the standard formula (Chap. V, § 3, n
◦ 13).
F is then said to be a primitive for f .
For simplicity’s sake, we will say that a function F is of class C
1/2 in I
if it is a primitive for a regulated function which we will always write F
; it
exists except possibly for a countable number of values of the variable : an
unimportant ambiguity that can be removed by setting F
(x) = F
d (x) for all
x. For this it is not sufficient that F be differentiable outside some countable
set. We adopt the notation C
1/2 because C
0 means that F is continuous, a
less restrictive condition, whereas C
1 means that F
exists everywhere and
is continuous, a more restrictive one.
2 Recall that a function f defined on an interval I ⊂ R is said to be regulated if it
satisfies the following three equivalent conditions : (a) it has both right and left
limits at all points of I, ; (b) for any compact interval K ⊂ I and r > 0, K can
be partitioned into intervals on which f is constant up to r; (c) there exists a
sequence of step functions converging uniformly to f on every compact set K ⊂ I
(hence on I if I is compact). Chap. V, n
◦ 7, Theorem 6. The Sum, product and
quotient of two regulated functions are also of the same type. If f and g are of
class C
1/2 , the product fg is a continuous function which admits a derivative
outside some countable set, the regulated function f
(t)g(t) + f (t)g
(t); fg is,
therefore, a primitive for f
g + fg
. This allows us to apply the integration by
parts formula to functions of class C
1/2 defined later.
3
§ 1. Integrals of Holomorphic Functions
1 – Preliminary Results
(i) The fundamental theorem (FT) of differential and integral calculus (Chap. V,
§ 3). In its simplest form, take a continuous function f on an interval I ⊂ R
and, choosing an a ∈ I, set
F (x) =
x
a
f (t)dt
a differentiable function such that F
(x) = f (x) for all x ∈ I is thus obtained.
Conversely, any primitive for f is given up to an additive constant by this
formula.
If we start with a regulated function
2 f – a less simple case –, the previous
formula defines a continuous function F admitting right-hand and left-hand
derivatives at every x ∈ I, given by
F
d (x) = lim
h = 0
h > 0
F (x + h) − F (x)
h
= f (x + 0) = lim
h = 0
h > 0
f (x + h)
and by a similar left-hand formula. In particular, the derivative F
(x) exists
outside some countable set D of discontinuous points of f . Conversely, if
there is regulated function f and a continuous function F in I which, outside
some countable subset of I, admits a derivative equal to f (x), then, up to
a constant, F is again given by the standard formula (Chap. V, § 3, n
◦ 13).
F is then said to be a primitive for f .
For simplicity’s sake, we will say that a function F is of class C
1/2 in I
if it is a primitive for a regulated function which we will always write F
; it
exists except possibly for a countable number of values of the variable : an
unimportant ambiguity that can be removed by setting F
(x) = F
d (x) for all
x. For this it is not sufficient that F be differentiable outside some countable
set. We adopt the notation C
1/2 because C
0 means that F is continuous, a
less restrictive condition, whereas C
1 means that F
exists everywhere and
is continuous, a more restrictive one.
2 Recall that a function f defined on an interval I ⊂ R is said to be regulated if it
satisfies the following three equivalent conditions : (a) it has both right and left
limits at all points of I, ; (b) for any compact interval K ⊂ I and r > 0, K can
be partitioned into intervals on which f is constant up to r; (c) there exists a
sequence of step functions converging uniformly to f on every compact set K ⊂ I
(hence on I if I is compact). Chap. V, n
◦ 7, Theorem 6. The Sum, product and
quotient of two regulated functions are also of the same type. If f and g are of
class C
1/2 , the product fg is a continuous function which admits a derivative
outside some countable set, the regulated function f
(t)g(t) + f (t)g
(t); fg is,
therefore, a primitive for f
g + fg
. This allows us to apply the integration by
parts formula to functions of class C
1/2 defined later.