116
VIII – Cauchy Theory
|f (h) − f (0+) − f
(0+)h| ≤ rh ;
the function f , extended at x = 0, therefore, has a derivative f
(0) = f
(0+).
A recurrence argument generalizes the result to successive derivatives.
As shown above, f and its successive derivatives are rapidly decreasing
functions at infinity. So f ∈ S(R + ).
(f) The Mellin inversion formula for ϕ. We now show that ϕ is indeed
the Mellin transform of f , i.e. that formula (17) defining f can be inverted.
Since it is merely a Fourier transform in disguise, it amounts to checking the
conditions used in section (i) for showing that the inversion formula applies
in both directions :
(a) the function ϕ is continuous on the vertical Re(s) = σ > 0,
(b) integral (17) is (absolutely) convergent,
(c)
|f (x)x
s
|d
∗ x < +∞ for Re(s) > 0;
the order of the conditions to be checked is changed as here we need to
calculate ϕ in terms of f and not f in terms of ϕ.
Checking (a) is trivial (ϕ is holomorphic), we would never have thought
of writing (17) if condition (b) was not satisfied, finally (c) is obvious since,
as seen above, f is continuous at x = 0 and, as noticed at the end of part (c)
of the proof, is a rapidly decreasing function at infinity.
The last assertion of the statement is totally unrelated to the Mellin
transform; apply the following result:
Lemma 2. Let ϕ be a function defined and holomorphic on an open set
U : a < Re(s) < b , Im(s) > c
and m be a real number. If the function s
m ϕ(s) is bounded at infinity on the
closed vertical strip of finite width contained in U , then the same holds for
all of ϕ’s derivatives.
To see this, argue as in n
◦ 4, (iv). Take a closed strip
B : Im(s) ≥ c
> c , a
≤ Re(s) ≤ b
with a < a
< b
< b
contained in U and choose some r > 0 such that
c < c
− r , a < a
− r , b
+ r < b .
The closed strip
B
: Im(s) ≥ c
− r , a − r ≤ Re(s) ≤ b + r
is contained in U and, for all s ∈ B, the disc centered at s and of radius r
is contained in B
. For s ∈ B, Cauchy’s formula (not quite correct, but the
forgotten numerical factor has no influence on the orders of magnitude)
VIII – Cauchy Theory
|f (h) − f (0+) − f
(0+)h| ≤ rh ;
the function f , extended at x = 0, therefore, has a derivative f
(0) = f
(0+).
A recurrence argument generalizes the result to successive derivatives.
As shown above, f and its successive derivatives are rapidly decreasing
functions at infinity. So f ∈ S(R + ).
(f) The Mellin inversion formula for ϕ. We now show that ϕ is indeed
the Mellin transform of f , i.e. that formula (17) defining f can be inverted.
Since it is merely a Fourier transform in disguise, it amounts to checking the
conditions used in section (i) for showing that the inversion formula applies
in both directions :
(a) the function ϕ is continuous on the vertical Re(s) = σ > 0,
(b) integral (17) is (absolutely) convergent,
(c)
|f (x)x
s
|d
∗ x < +∞ for Re(s) > 0;
the order of the conditions to be checked is changed as here we need to
calculate ϕ in terms of f and not f in terms of ϕ.
Checking (a) is trivial (ϕ is holomorphic), we would never have thought
of writing (17) if condition (b) was not satisfied, finally (c) is obvious since,
as seen above, f is continuous at x = 0 and, as noticed at the end of part (c)
of the proof, is a rapidly decreasing function at infinity.
The last assertion of the statement is totally unrelated to the Mellin
transform; apply the following result:
Lemma 2. Let ϕ be a function defined and holomorphic on an open set
U : a < Re(s) < b , Im(s) > c
and m be a real number. If the function s
m ϕ(s) is bounded at infinity on the
closed vertical strip of finite width contained in U , then the same holds for
all of ϕ’s derivatives.
To see this, argue as in n
◦ 4, (iv). Take a closed strip
B : Im(s) ≥ c
> c , a
≤ Re(s) ≤ b
with a < a
< b
< b
contained in U and choose some r > 0 such that
c < c
− r , a < a
− r , b
+ r < b .
The closed strip
B
: Im(s) ≥ c
− r , a − r ≤ Re(s) ≤ b + r
is contained in U and, for all s ∈ B, the disc centered at s and of radius r
is contained in B
. For s ∈ B, Cauchy’s formula (not quite correct, but the
forgotten numerical factor has no influence on the orders of magnitude)