118
VIII – Cauchy Theory
theorem 14, it follows that the function s
n ϕ(s) is bounded at infinity (hence
so are its successive derivatives) on any vertical strip of finite width.
We leave it to the reader to show that conversely any meromorphic function ϕ(s) on C with these two properties (its only singularities are simple
poles at points −u 0 > −u 1 > . . . on the real axis and it is rapidly decreasing
at infinity on any vertical strip of finite width) is the Mellin transform of a
function of the previous type.
14 – Stirling’s Formula for the Gamma Function
The simplest example of an application of the inversion formula can be obtained by choosing f (x) = e
−x , obviously in S(R + ). Its Mellin transform is,
by definition, Γ (s). Hence
e
−x =
1
2πi
Re(s)=σ
Γ (s)x
−s ds =
1
2π
Γ (σ + it)x
−σ−it dt , σ > 0 ,
(14.1)
and a slightly less simple formula for −p − 1 < σ < −p. This result is due
to Mellin himself (1910), but it can be easily obtained without invoking the
general theorem: it suffices to reconstruct the proof in this particular case. . .
Theorem 14 also shows that, on any vertical not passing through a pole,
the function t → Γ (σ + it) is in the Schwartz space. This is a weak result:
indeed, formula (27) that will be proved at the end of this section shows that
the Γ function decreases exponentially on every vertical.
This result is based on an evaluation (Stieltjes) which for s ∈ N reduces
to Stirling’s formula
n! ∼ (2π)
1/2 n
n+1/2 e
−n
as n −→ +∞
(14.2)
of Chapter VI, n
◦ 18, namely
Γ (s) ∼ (2π)
1/2 s
s−1/2 e
−s
(14.3)
as s tends to infinity in an angle | Arg(s)| ≥ π − δ with δ > 0; for integer s ,
the equivalence with (2) follows from the relation
(n − 1)! = n!/n ∼ (2π)
1/2 n
n−1/2 e
−n .
We first prove
64 relation (3), then we show how to deduced the behaviour of
the Γ function on the verticals. These results arise in fields such as analytic
number theory and in the study of the asymptotic behaviour of important
64 The rest of this n
◦ is essentially a fairly concise, detailed presentation of Remmert 2, Chap. 2, § 4. Dieudonn´ e, Calcul infinit´ esimal, IX.7.6, gives a genuine
asymptotic expansion by using in full the Euler-MacLaurin formula. N. Bourbaki, Fonctions d’une variable r´ eelle is another reference.
VIII – Cauchy Theory
theorem 14, it follows that the function s
n ϕ(s) is bounded at infinity (hence
so are its successive derivatives) on any vertical strip of finite width.
We leave it to the reader to show that conversely any meromorphic function ϕ(s) on C with these two properties (its only singularities are simple
poles at points −u 0 > −u 1 > . . . on the real axis and it is rapidly decreasing
at infinity on any vertical strip of finite width) is the Mellin transform of a
function of the previous type.
14 – Stirling’s Formula for the Gamma Function
The simplest example of an application of the inversion formula can be obtained by choosing f (x) = e
−x , obviously in S(R + ). Its Mellin transform is,
by definition, Γ (s). Hence
e
−x =
1
2πi
Re(s)=σ
Γ (s)x
−s ds =
1
2π
Γ (σ + it)x
−σ−it dt , σ > 0 ,
(14.1)
and a slightly less simple formula for −p − 1 < σ < −p. This result is due
to Mellin himself (1910), but it can be easily obtained without invoking the
general theorem: it suffices to reconstruct the proof in this particular case. . .
Theorem 14 also shows that, on any vertical not passing through a pole,
the function t → Γ (σ + it) is in the Schwartz space. This is a weak result:
indeed, formula (27) that will be proved at the end of this section shows that
the Γ function decreases exponentially on every vertical.
This result is based on an evaluation (Stieltjes) which for s ∈ N reduces
to Stirling’s formula
n! ∼ (2π)
1/2 n
n+1/2 e
−n
as n −→ +∞
(14.2)
of Chapter VI, n
◦ 18, namely
Γ (s) ∼ (2π)
1/2 s
s−1/2 e
−s
(14.3)
as s tends to infinity in an angle | Arg(s)| ≥ π − δ with δ > 0; for integer s ,
the equivalence with (2) follows from the relation
(n − 1)! = n!/n ∼ (2π)
1/2 n
n−1/2 e
−n .
We first prove
64 relation (3), then we show how to deduced the behaviour of
the Γ function on the verticals. These results arise in fields such as analytic
number theory and in the study of the asymptotic behaviour of important
64 The rest of this n
◦ is essentially a fairly concise, detailed presentation of Remmert 2, Chap. 2, § 4. Dieudonn´ e, Calcul infinit´ esimal, IX.7.6, gives a genuine
asymptotic expansion by using in full the Euler-MacLaurin formula. N. Bourbaki, Fonctions d’une variable r´ eelle is another reference.