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.