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.
Previous Page Next Page