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.