§ 2. Cauchy’s Integral Formulas
59
the function μ
(t) is regulated, but the general case is not hard since, in all
questions of this type, the only properties of measures really used are their
definitions – linearity of f → μ(f ) and upper bound in terms of the uniform
norm of f – and elementary theorems about passing to the limit under the
sign that follow directly from the definition.
7 – Integrals Depending Holomorphically on a Parameter
Theorem 9. Let I be an interval, μ a measure on I, U an open subset of C
and f : I × U −→ C a function satisfying the following conditions :
(a) f is continuous on I × U ,
(b) f (t, z) is a holomorphic function of z for all t ∈ I,
(c) for any compact set H ⊂ U , there exists a positive μ-integrable
39 function p H (t) on I such that
|f (t, z)| ≤ p H (t) for all t ∈ I and z ∈ H .
(7.1)
Let f
(r) (t, z) be the derivative of order r of z → f (t, z). Then f
(r) (t, z) satisfies conditions (a), (b) and (c) for all r, the function
g(z) =
f (t, z)dμ(t)
(7.2)
is holomorphic on U , the functions f
(r) (t, z) are μ-integrable and
g
(r) (z) =
f
(r) (t, z)dμ(t) for all r ∈ N .
(7.3)
It is sufficient to prove the statements with respect to r = 1 : the general
case will follow by a repeated application of the result.
First proof. Theorem 24 bis of Chap. V, § 7 is analogous to the result
we need to prove but based on different assumptions : f
(instead of f ) was
assumed to satisfy conditions (a) and (c). At that point, the only tool at
our disposal was indeed the formula for differentiation of an integral with
respect to a real parameter; it assumes that the derivative being integrated is
continuous and that its integral converges normally on compact sets. We then
obtained the holomorphy of (1) and formula (2) by differentiating integral (1)
with respect to coordinates x and y of z and by checking Cauchy’s condition
39 If dμ(t) = μ
(t)dt with μ
(t) regulated, this means that
pH (t)|μ
(t)|dt < +∞.
In the case of an arbitrary positive measure, pH (t) can be assumed to be lsc (and
in practice, even continuous) because a (Lebesgue) integrable positive function is
always dominated by an integrable lsc function. Besides, note that (c) is always
satisfied when I is compact because f is bounded on the compact set I × H, so
that it suffices to choose the constant function pH (t) = sup |f (s, z)|, sup being
extended to (s, z) ∈ I × H.
59
the function μ
(t) is regulated, but the general case is not hard since, in all
questions of this type, the only properties of measures really used are their
definitions – linearity of f → μ(f ) and upper bound in terms of the uniform
norm of f – and elementary theorems about passing to the limit under the
sign that follow directly from the definition.
7 – Integrals Depending Holomorphically on a Parameter
Theorem 9. Let I be an interval, μ a measure on I, U an open subset of C
and f : I × U −→ C a function satisfying the following conditions :
(a) f is continuous on I × U ,
(b) f (t, z) is a holomorphic function of z for all t ∈ I,
(c) for any compact set H ⊂ U , there exists a positive μ-integrable
39 function p H (t) on I such that
|f (t, z)| ≤ p H (t) for all t ∈ I and z ∈ H .
(7.1)
Let f
(r) (t, z) be the derivative of order r of z → f (t, z). Then f
(r) (t, z) satisfies conditions (a), (b) and (c) for all r, the function
g(z) =
f (t, z)dμ(t)
(7.2)
is holomorphic on U , the functions f
(r) (t, z) are μ-integrable and
g
(r) (z) =
f
(r) (t, z)dμ(t) for all r ∈ N .
(7.3)
It is sufficient to prove the statements with respect to r = 1 : the general
case will follow by a repeated application of the result.
First proof. Theorem 24 bis of Chap. V, § 7 is analogous to the result
we need to prove but based on different assumptions : f
(instead of f ) was
assumed to satisfy conditions (a) and (c). At that point, the only tool at
our disposal was indeed the formula for differentiation of an integral with
respect to a real parameter; it assumes that the derivative being integrated is
continuous and that its integral converges normally on compact sets. We then
obtained the holomorphy of (1) and formula (2) by differentiating integral (1)
with respect to coordinates x and y of z and by checking Cauchy’s condition
39 If dμ(t) = μ
(t)dt with μ
(t) regulated, this means that
pH (t)|μ
(t)|dt < +∞.
In the case of an arbitrary positive measure, pH (t) can be assumed to be lsc (and
in practice, even continuous) because a (Lebesgue) integrable positive function is
always dominated by an integrable lsc function. Besides, note that (c) is always
satisfied when I is compact because f is bounded on the compact set I × H, so
that it suffices to choose the constant function pH (t) = sup |f (s, z)|, sup being
extended to (s, z) ∈ I × H.