450
D Invariant Measures
where dd H 1 is the invariant measure associated with the Hermitian matrix ¯
H 1 (see
Eq. (D.21)). In Eq. (D.103), we have used the fact that
∞
−∞
dt
(at 2 + 2bt + c) α =
√
π
−
1
2 )
(α)
(ac − b 2 )
1
2 −α
a 1−α
.
(D.104)
The integrand in Eq. (D.103) can be simplified following the analysis described in
Hua (1963). Let us introduce a new column vector, ¯
w, defined as
¯
T 1 · ¯
v = ¯
w≡
⎛
⎜
⎜
⎜
⎝
w 1
w 2
. . .
w N −1
⎞
⎟
⎟
⎟
⎠
,
(D.105)
where the matrix elements, w j (j = 1, 2, . . ., N − 1), are complex numbers. After
some matrix algebra, Eq. (D.99) can be written
A = 1 − ¯
w
†
·
1
( ¯
1 + ¯
w· ¯
w † )
· ¯
w =
1
1 + ¯
w † · ¯
w
.
(D.106)
To obtain the expression in Eq. (D.106), we have used the expansion
1
( ¯
1 + ¯
w· ¯
w † )
= ¯
1 − ¯
w· ¯
w
†
+ ¯
w· ¯
w
†
· ¯
w· ¯
w
†
− . . .,
(D.107)
and then we rearrange terms. Similarly, Eq. (D.100) takes the form
B = − ¯
w
†
·
1
( ¯
1 + ¯
w· ¯
w † )
· ¯
H 1 · ¯
w = −
( ¯
w † · ¯
H 1 · ¯
w)
(1 + ¯
w † · ¯
w)
,
(D.108)
and after some algebra, Eq. (D.101) takes the form
C = 1 + ¯
w
†
· ¯
w +
( ¯
w † · ¯
H 1 · ¯
w) 2
(1 + ¯
w † · ¯
w)
.
(D.109)
If we multiply Eqs. (D.106), (D.108), and (D.109) together, we find that
AC − B
2
= 1.
(D.110)
Thus Eq. (D.103) can be written
I
α
N (2) =
√
π
−
1
2 )
(α)
2
N −1