124
VIII – Cauchy Theory
|s + x|
2
≥ (r + x)
2 cos
2 ϕ/2 .
Since the periodic function P 2 (x) is contained between 0 and 1/8 everywhere,
8|μ(s)| cos
2 ϕ/2 ≤
+∞
0
(r + x)
−2 dx = 1/r
or (Remmert, p. 52)
|μ(s)| ≤ 1/8|s| cos
2 ϕ/2 in C + .
(14.26)
Hence lim μ(s) = 0 if s tends to infinity in such a way that the product
|s| cos
2 ϕ/2 does so as well, for example if we consider a subset of C defined
by
| Arg(s)| ≤ π − δ with 0 < δ < π .
The Stieltjes formula holds under this condition, for example if s remains in
the half-plane Re(s) > c and hence in a vertical strip of finite width.
As for Remmert, he avoids all the limit calculations that have been detailed here. He a priori introduces the function μ(s) and using the second
integral, he notices through an elementary calculation that
μ(s) − μ(s + 1) = (s + 1/2) Log(1 + 1/s) − 1 ,
then that the function f (s) = s
s−1/2 e
−s e
μ(s) , holomorphic on C + , satisfies
Wielandt’s assumptions [n
◦ 10, (i)]. Hence f (s) = f (1)Γ (s), and in particular
f (n) = f (1)(n − 1)!; as μ(n) obviously tends to 0, comparison with Stirling’s
formula shows thatf (1) = (2π)
1/2 . An excellent example of Blitzbeweis !
To show that the gamma function decreases exponentially on the verticals,
|s
s−1/2
| = e
Re[(s−1/2) Log s]
still needs to be evaluated For s = σ + it,
Re [(s − 1/2) Log s] = (σ − 1/2) log |s| − t Arg(s) .
As s tends to infinity in the vertical strip B of finite width, the argument of
s tends to π/2 if t tends to +∞, and to −π/2 if t tends to −∞; in the first
case, using the power series for Arctg,
π/2 − Arg(s) = Arctg(σ/t) = σ/t + O(t
−3 )
since σ remains in a compact set, and so
−t Arg(s) = −π|t|/2 + σ + O(t
−2 ) ,
a result which also holds in the second case. For the same reason,
VIII – Cauchy Theory
|s + x|
2
≥ (r + x)
2 cos
2 ϕ/2 .
Since the periodic function P 2 (x) is contained between 0 and 1/8 everywhere,
8|μ(s)| cos
2 ϕ/2 ≤
+∞
0
(r + x)
−2 dx = 1/r
or (Remmert, p. 52)
|μ(s)| ≤ 1/8|s| cos
2 ϕ/2 in C + .
(14.26)
Hence lim μ(s) = 0 if s tends to infinity in such a way that the product
|s| cos
2 ϕ/2 does so as well, for example if we consider a subset of C defined
by
| Arg(s)| ≤ π − δ with 0 < δ < π .
The Stieltjes formula holds under this condition, for example if s remains in
the half-plane Re(s) > c and hence in a vertical strip of finite width.
As for Remmert, he avoids all the limit calculations that have been detailed here. He a priori introduces the function μ(s) and using the second
integral, he notices through an elementary calculation that
μ(s) − μ(s + 1) = (s + 1/2) Log(1 + 1/s) − 1 ,
then that the function f (s) = s
s−1/2 e
−s e
μ(s) , holomorphic on C + , satisfies
Wielandt’s assumptions [n
◦ 10, (i)]. Hence f (s) = f (1)Γ (s), and in particular
f (n) = f (1)(n − 1)!; as μ(n) obviously tends to 0, comparison with Stirling’s
formula shows thatf (1) = (2π)
1/2 . An excellent example of Blitzbeweis !
To show that the gamma function decreases exponentially on the verticals,
|s
s−1/2
| = e
Re[(s−1/2) Log s]
still needs to be evaluated For s = σ + it,
Re [(s − 1/2) Log s] = (σ − 1/2) log |s| − t Arg(s) .
As s tends to infinity in the vertical strip B of finite width, the argument of
s tends to π/2 if t tends to +∞, and to −π/2 if t tends to −∞; in the first
case, using the power series for Arctg,
π/2 − Arg(s) = Arctg(σ/t) = σ/t + O(t
−3 )
since σ remains in a compact set, and so
−t Arg(s) = −π|t|/2 + σ + O(t
−2 ) ,
a result which also holds in the second case. For the same reason,