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 ˜ a for any complex value a which, when applied to a Schwartz function φ, reads (1.17) ˜ 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 ˜ 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 ˜ −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 ˜ −k) at nonpositive integers: ˜ Γ(0) := ΓHad(0) = ΓRiesz(0) = −γ if k = 0, ˜ −k) := ΓHad(−k) = ΓRiesz(−k) = (−1)k k! k j=1 1 j γ⎦ if k 0. Extending a homogeneous distribution Fa ˜ 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.
Previous Page Next Page