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Numerical Integration 63
(3.10.5) ˜ω (n)
j
= Γ(n + α + 1)
4n 2 n!
4n + η (n)
j
ˆL (α)
n (η (n)
j )
where we noted that, for n ≥ 2
(3.10.6)
Again we have
(3.10.7)
where ˆp := pS (α)
n .
d
S
dx
(α)
n−1 (x)
−1
:= ˜w (n)
j
S (α)
n (η (n)
−2 j )
d
dx ˆ L (α)
n−1 (η(n) j ) + ˆ L (α)
n−1 (η(n)
n−1
j )
m=1
=
+∞
p
0
2 (x) x α e −x dx =
S (α)
n−1 (x)
−1 n−1
n−1
j=0
m=1
1
4m + η (n)
j
1
, x ∈ [0,+∞[.
4m + x
ˆp 2 (η (n)
j ) ˜ω (n)
j , ∀p ∈ Pn−1,
The same procedure applied to the Hermite weights in (3.4.9), yields
(3.10.8) ω (n)
j
=
√ π n!
2 n−1 [(n/2)!] 2
(3.10.9) ω (n)
j
=
:= w (n)
j
S (−1/2)
n/2 ([ξ (n)
j ] 2 −2 )
ˆH ′
n(ξ (n)
−2 j ) , 1 ≤ j ≤ n, if n is even,
:= w (n)
j
√ π n!
2 n−1 [((n − 1)/2)!] 2
S (1/2)
(n−1)/2 ([ξ(n) j ] 2 −2 )
ˆH ′
n(ξ (n)
−2 j ) , 1 ≤ j ≤ n, if n is odd.
−1
1 ≤ j ≤ n − 1,
Here the ξ (n)
j ’s are the Hermite zeroes. Relations (1.7.9), (1.7.10), (1.7.11) and (1.7.12)
have been taken into account.
,