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( ).
