12 1. GAMMA FUNCTION EXTENDED TO NONPOSITIVE INTEGER POINTS
(3) Show that if b = −k −N {0}, the map

x−k−1 e−x
dx lies
in A[ ]
−k+1
A[ ] log . The corresponding asymptotic expansion has
constant term given by
ΓHad(0)
= −γ if k = 0,
ΓHad(−k)
=
(−1)k
k!


k
j=1
1
j

γ⎦

,
where the sum over j is set to zero when k = 1.
1.5. Discrepancies
The following theorem sums up the results of the two previous paragraphs.
Theorem 1.24. The Riesz and Hadamard regularisation methods yield the
same extended distribution
˜
F
a
for any complex value a which, when applied to a
Schwartz function φ, reads
(1.17)
˜
F
a
(φ) :=
∞,Riesz
0
xa
φ(x) dx =
∞,Had
0
xa
φ(x) dx.
These coincide with the ordinary integral

0
xa φ(x) dx whenever Re(a) −1.
(1) If a / −N, then
˜
F a(φ) =
(−1)k
(a + 1) · · · (a + k)
Fa+k(φ(k)),
where k is any integer such that Re(a + k) −1.
(2) Furthermore, for a positive integer k
˜
F −k(φ) =
φ(k−1)(0)
(k 1)!
k−1
j=1
1
j

1
(k 1)!

0
log x
φ(k)(x)dx,
setting the sum over j equal to zero if k = 1.
This applied to the Schwartz function φ(x) =
e−x
confirms the results of Exer-
cises 1.23 and 1.18, which show that Riesz and Hadamard finite part regularisations
lead to the same extended Gamma function
˜
Γ( −k) at nonpositive integers:
˜
Γ(0) :=
ΓHad(0)
=
ΓRiesz(0)
= −γ if k = 0,
˜
Γ( −k) :=
ΓHad(−k)
=
ΓRiesz(−k)
=
(−1)k
k!


k
j=1
1
j

γ⎦

if k 0.
Extending a homogeneous distribution Fa
˜
F
a
to negative integers unfortunately
has a cost for we lose various properties along the way.
1.5.1. Loss of homogeneity. Recall from (1.1) that Fa is positively homo-
geneous for Re(a) −1:
Fa(φt) =
ta
Fa(φ) ∀t 0.
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