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. 4 I thank Bing Zhang for interesting discussions and comments concerning this paragraph and Florian Hanisch for his very constructive suggestions. 5 That 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.
Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2012 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.