8 1. GAMMA FUNCTION EXTENDED TO NONPOSITIVE INTEGER POINTS
Let us introduce some
notation.4
A[ ] denotes the set of germs of smooth
functions around zero in the variable
,5
and for any complex number α and any
positive integer l we set
A[ ]
α
:= {f( )
α,
f ∈ A[ ]},
A[ ]
logl
:= {f( )
logl
, f ∈ A[ ]}. (1.15)
Using a Taylor expansion of f at zero, a function g( ) = f( )
α
in A[ ]
α
reads
g( ) =
N
k=0
f
k(0)
k!
α+k
+ o(
N+α)
∀N ∈ N.
The finite part is given by the constant term in the expansion
fp
=0
g( ) =
N
k=0
f
k(0)
k!
δα+k,
independently of the choice of N provided it is chosen large enough. Similarly,
for a function h( ) = f( )
logl
in A[ ]
logl
for some positive integer l, we set
fp
=0
h( ) = 0. We need a technical lemma.
Lemma 1.19. Let φ denote a Schwartz function on R+. The map −→
∞
log x φ(x) dx lies in A[ ] A[ ] log and
lim
→0
∞
log x φ(x) dx =
∞
0
log x φ(x) dx.
Proof. We have
∞
log x φ(x) dx =
∞
0
log x φ(x) dx −
0
log x φ(x) dx
=
∞
0
log x φ(x) dx −
1
0
log( x) φ( x) dx
=
∞
0
log x φ(x) dx −
1
0
log x φ( x) dx
− log
1
0
φ( x) dx.
A Taylor expansion of φ at 0 shows that →
1
0
φ(x) dx lies in A[ ]. The map
1
0
log x φ(x) dx also lies in A[ ]; indeed
1
0
log x φ(x) dx = −
1
0
(x log x − x) φ ( x) dx − φ( )
and a Taylor expansion of φ and φ at 0 provides the required asymptotic expansion.
Hence the map →
∞
log x φ(x) dx lies in A[ ] A[ ] log , and we have
lim
→0
∞
log x φ(x) dx =
∞
0
log x φ(x) dx.
4I
thank Bing Zhang for interesting discussions and comments concerning this paragraph and
Florian Hanisch for his very constructive suggestions.
5That
is, equivalence classes of smooth functions are defined on a neighborhood of zero for
the equivalence relation f ∼ g if f and g coincide on some open neighborhood of zero.
