§ 3. Some Applications of Cauchy’s Method
79
ˆ
ϕ(y) = w
−s
+∞
0
e
−x x
s d
∗ x = Γ (s)w
−s
(10.4)
and so (1) would follow. The analyticity with respect to w remains to be
proved.
(i) The gamma function. Some of the many properties
47 of this function
are already known, to begin with the formulas
Γ (s + 1) = sΓ (s), Γ(n) = (n − 1)! for n ≥ 1 .
(10.5.1)
This relation immediately shows that Γ (s) can be extended analytically to
all of C, excepting the simple poles at s = 0, −1, . . ., where
Res(Γ, −n) = (−1)
n /n!
(10.5.2)
(Chapter V, § 7, n
◦ 25, Example 5); in fact,
Γ (s) =
1
0
e
−x x
s d
∗ x +
+∞
1
e
−x x
s d
∗ x =
(10.5.3)
=
N
(−1)
n /n!(s + n) + Γ
+ (s).
The integral over (1, +∞), Γ
+ (s), converges for all s and is an entire function.
Contrary to the integral over (0, 1), the series obtained by integrating term
by term the exponential series is also convergent for all s.
Setting
f n (x) =
(1 − x/n)
n x
s for x ≤ n ,
0
f o r x > n ,
we get a sequence of functions converging to e
−x x
s while remaining dominated by e
−x x
s (exercise !); hence
Γ (s) = lim
f n (x)d
∗ x = lim n!n
s /s(s + 1) . . . (s + n)
(10.5.4)
(Chapter V, § 7, n
◦ 23, Example 1), a priori for Re(s) > 0. This leads to the
expansion
1/Γ (s) = se
Cs
(1 + s/n)e
−s/n
(10.5.5)
47 See for example Dieudonn´ e, Calcul infinit´ esimal (Hermann, 1968), IV.3, IX-4
to IX-8, Remmert, Funktionentheorie 2, Chap. 2, § 2, where the function is defined by using its infinite product, Freitag and Busam, Funktionentheorie, chap.
IV, § 1 and in particular the exercises, not to mention earlier authors. Entire
books have been written on it, notably N. Nielsen, Handbuch der Theorie der
Gammafunktion (Leipzig, 1906, reedit. Chelsea, 1965).
79
ˆ
ϕ(y) = w
−s
+∞
0
e
−x x
s d
∗ x = Γ (s)w
−s
(10.4)
and so (1) would follow. The analyticity with respect to w remains to be
proved.
(i) The gamma function. Some of the many properties
47 of this function
are already known, to begin with the formulas
Γ (s + 1) = sΓ (s), Γ(n) = (n − 1)! for n ≥ 1 .
(10.5.1)
This relation immediately shows that Γ (s) can be extended analytically to
all of C, excepting the simple poles at s = 0, −1, . . ., where
Res(Γ, −n) = (−1)
n /n!
(10.5.2)
(Chapter V, § 7, n
◦ 25, Example 5); in fact,
Γ (s) =
1
0
e
−x x
s d
∗ x +
+∞
1
e
−x x
s d
∗ x =
(10.5.3)
=
N
(−1)
n /n!(s + n) + Γ
+ (s).
The integral over (1, +∞), Γ
+ (s), converges for all s and is an entire function.
Contrary to the integral over (0, 1), the series obtained by integrating term
by term the exponential series is also convergent for all s.
Setting
f n (x) =
(1 − x/n)
n x
s for x ≤ n ,
0
f o r x > n ,
we get a sequence of functions converging to e
−x x
s while remaining dominated by e
−x x
s (exercise !); hence
Γ (s) = lim
f n (x)d
∗ x = lim n!n
s /s(s + 1) . . . (s + n)
(10.5.4)
(Chapter V, § 7, n
◦ 23, Example 1), a priori for Re(s) > 0. This leads to the
expansion
1/Γ (s) = se
Cs
(1 + s/n)e
−s/n
(10.5.5)
47 See for example Dieudonn´ e, Calcul infinit´ esimal (Hermann, 1968), IV.3, IX-4
to IX-8, Remmert, Funktionentheorie 2, Chap. 2, § 2, where the function is defined by using its infinite product, Freitag and Busam, Funktionentheorie, chap.
IV, § 1 and in particular the exercises, not to mention earlier authors. Entire
books have been written on it, notably N. Nielsen, Handbuch der Theorie der
Gammafunktion (Leipzig, 1906, reedit. Chelsea, 1965).