104
VIII – Cauchy Theory
Γ f (s) =
+∞
0
f (x)x
s d
∗ x ,
(13.1)
where x
s = exp(s log x) is the real, positive function of Chap. IV for real s, to
“ any ” function
61 f (x) defined for x > 0. The notation recalls the fact that
Γ f (s) = Γ (s) if f (x) = e
−x .
(13.2)
As Γ f (is) is the complex Fourier transform of t → 2πf (e
2πt ), the general
statements of n
◦ 12, (i) are easily translated.
In conformity to a long tradition, setting
s = σ + it ,
absolute convergence of (1) only depends on Re(s) = σ. Clearly, Γ f is defined
on a strip of the plane of the form Re(s) ∈ I, where I is a priori an arbitrary
interval; it is holomorphic on the interior of this strip and bounded on any
closed vertical strip of finite width where it is defined. Theorem 9 of n
◦ 7
shows that its derivatives are given by
Γ
(n)
f (s) =
+∞
0
f (x) log
n x.x
s d
∗ x
(13.3)
for all n ∈ N. The logarithmic factor does not destroy convergence: since we
are in the interior of the convergence strip, the integral
f (x)x
s d
∗ x indeed
converges at s = a and s = b, where a < Re(s) < b, and it is sufficient to
observe that
(log x)
n x
s = o(x
a ) as x tends to 0 ,
(log x)
n x
s = o(x
b ) as x tends to +∞
to justify the result, which is anyhow guaranteed by theorem 9 of n
◦ 7.
Convergence of the Mellin integral in the neighbourhood of 0 is ensured
for Re(s) > 0 if f is bounded in the neighbourhood of 0, though it converges
at infinity if f is integrable at infinity with respect to dx and if the function
x
s−1 is bounded at infinity, i.e. for Re(s) < 1. The strip where the function Γ f
is defined is, therefore, at least 0 < Re(s) < 1 if f is bounded and integrable
with respect to x over R + .
As Γ f (is) is the value of the complex Fourier transform g(z) of 2πf (e
2πu ) =
ˆ
g(u) at z = is, and as there is sometimes an inversion formula
ˆ
g(u) =
Re(z)=y
g(z)e(−uz)dz ,
in horizontal strip where g is defined, a similar result on Mellin transforms
will presumably follow by making suitable assumptions.; variable changes
61 In practice, f (x) is regulated and almost always continuous for x > 0.
VIII – Cauchy Theory
Γ f (s) =
+∞
0
f (x)x
s d
∗ x ,
(13.1)
where x
s = exp(s log x) is the real, positive function of Chap. IV for real s, to
“ any ” function
61 f (x) defined for x > 0. The notation recalls the fact that
Γ f (s) = Γ (s) if f (x) = e
−x .
(13.2)
As Γ f (is) is the complex Fourier transform of t → 2πf (e
2πt ), the general
statements of n
◦ 12, (i) are easily translated.
In conformity to a long tradition, setting
s = σ + it ,
absolute convergence of (1) only depends on Re(s) = σ. Clearly, Γ f is defined
on a strip of the plane of the form Re(s) ∈ I, where I is a priori an arbitrary
interval; it is holomorphic on the interior of this strip and bounded on any
closed vertical strip of finite width where it is defined. Theorem 9 of n
◦ 7
shows that its derivatives are given by
Γ
(n)
f (s) =
+∞
0
f (x) log
n x.x
s d
∗ x
(13.3)
for all n ∈ N. The logarithmic factor does not destroy convergence: since we
are in the interior of the convergence strip, the integral
f (x)x
s d
∗ x indeed
converges at s = a and s = b, where a < Re(s) < b, and it is sufficient to
observe that
(log x)
n x
s = o(x
a ) as x tends to 0 ,
(log x)
n x
s = o(x
b ) as x tends to +∞
to justify the result, which is anyhow guaranteed by theorem 9 of n
◦ 7.
Convergence of the Mellin integral in the neighbourhood of 0 is ensured
for Re(s) > 0 if f is bounded in the neighbourhood of 0, though it converges
at infinity if f is integrable at infinity with respect to dx and if the function
x
s−1 is bounded at infinity, i.e. for Re(s) < 1. The strip where the function Γ f
is defined is, therefore, at least 0 < Re(s) < 1 if f is bounded and integrable
with respect to x over R + .
As Γ f (is) is the value of the complex Fourier transform g(z) of 2πf (e
2πu ) =
ˆ
g(u) at z = is, and as there is sometimes an inversion formula
ˆ
g(u) =
Re(z)=y
g(z)e(−uz)dz ,
in horizontal strip where g is defined, a similar result on Mellin transforms
will presumably follow by making suitable assumptions.; variable changes
61 In practice, f (x) is regulated and almost always continuous for x > 0.