80
VIII – Cauchy Theory
of the function Γ (s) into an infinite product convergent everywhere, and so
valid everywhere by analytic extension;
C = lim(1 + . . . + 1/n − log n) = 0, 577215664 . . .
is the Euler constant (Chapter VI, § 2, n
◦ 18). The complement formula (same
reference)
Γ (s)Γ (1 − s) = π/sinπs
(10.5.6)
is, for example, obtained by comparing the infinite product expansions of
both sides with that of
sin πs = πs
n≥1
1 − s
2 /n
2
(Chap. IV, n
◦ 18). It follows that
Γ (1/2) = π
1/2 ,
(10.5.7)
a result that reduces to the integral of exp(−πx
2 ), and that
|Γ (1/2 + it)|
2 = π/ cosh πt for t ∈ R .
(10.5.8)
The duplication formula
Γ (2s) = π
−1/2 2
2s−1 Γ (s)Γ (s + 1/2)
(10.5.9)
which is often used in analytic number theory will be needed. A very (too?)
ingenious method
48 for obtaining it consists in using a characterization due
to Helmut Wielandt (1939) of the gamma function by the following two properties :
(a) f is holomorphic on a domain G containing the strip 1 ≤ Re(z) ≤ 2 and
bounded on it;
(b) f (s + 1) = sf (s) whenever s, s + 1 ∈ G.
Property (b) allows us, as in the case of Euler’s function, to first find an
analytically extension of f to all of C with, at worst, simple poles at integers
≤ 0 and
Res(f, −n) = (−1)
n f (1)/n! .
As a result, g(s) = f (s) − f (1)Γ (s) is an entire function also satisfying g(s +
1) = sg(s). The entire function h(s) = g(s)g(1 − s) then satisfies h(s + 1) =
−h(s).
48 I find it in Freitag-Busam, Chap. IV, § 1 and in Remmert 2, Chap. 2, § 2. A
more general, but far less useful, formula can be found in Dieudonn´ e, Calcul
infinit´ esimal, IX.4.