10 1. GAMMA FUNCTION EXTENDED TO NONPOSITIVE INTEGER POINTS For Re(a) ∈ ] − (k + 1), −k] and a / −N, using a Taylor expansion of φ(l) at 0, we have ∞ xa φ(x) dx = (−1)k (a + 1) · · · (a + k) ∞ xa+k φ(k)(x) dx + (−1)k a+k (a + 1) · · · (a + k) φ(k−1)( ) + · · · + (−1)j a+j (a + 1) · · · (a + j) φ(j−1)( ) + · · · − a+1 a + 1 φ( ) = (−1)k (a + 1) · · · (a + k) ∞ xa+k φ(k)(x) dx + k j=1 (−1)j Nj ij=0 a+j+ij (a + 1) · · · (a + j) ij! φ(j−1+ij)(0) + remainder terms. The constant term reads ∞,Had 0 xa φ(x) dx = (−1)k (a + 1) · · · (a + k) ∞ 0 xa+k φ(k)(x) dx since the remaining terms do not contribute to the constant term. (3) If a = −1 then, ∞ x−1 φ(x) dx = − ∞ log x φ (x) dx + [log x φ(x)]∞ = − ∞ log x φ (x) dx − log φ( ), which lies in A[ ] A[ ] log as a consequence of Lemma 1.19 and has finite part at zero given by ∞,Had 0 x−1 φ(x) dx = − ∞ 0 log x φ (x) dx. (4) If a = −k for some integer k 1, then by induction on k we show that the map → ∞ x−k φ(x) dx lies in A[ ] −k+1 A[ ] log . The previous step gives the statement for k = 1. Using integration by parts, we easily prove the induction step, ∞ x−k φ(x) dx = − ∞ x−k+1 −k + 1 φ (x) dx + x−k+1 −k + 1 φ(x) ∞ = 1 k − 1 ∞ x−k+1 φ (x) dx + −k+1 k − 1 φ( ).
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