Spatial Characterization Of Two-Photon States - GAP-Optique

B. Integrals of the matrix mode function
where the integrals in each variable are independent, so it is possible to write
them as
dx exp − xtAx
=
2 + ixt
b
dy1 exp − y2 1d1
2 + iy1b ′ 1
dy2
− y2 2d2
2 + iy2b ′
2 ...
which, after solving each integral separately, becomes
dx exp − xtAx 2 + ixt
b =
n
j=1
2π
dj
exp
dyn
− b′ 2
j
2dj
− y2 ndn
2 + iynb ′
n ,
(B.5)
. (B.6)
Because D is a diagonal matrix, 1/dj are the elements of D−1 , and n j=1 dj =
det(D); thus it is possible to write the last expression as
dx exp − xtAx 2 + ixt
b = (2π)n/2
exp
det(D)
− b′ t D −1 b ′
2
. (B.7)
Finally, since det(A) = det(D) and b ′ t D −1 b ′ = b t A −1 b, the integral becomes
dx exp − xtAx 2 + ixt
b = (2π)n/2
exp −
det(A) btA−1
b
. (B.8)
2
This proof can be applied to the special case in which all the elements of the
vector b are equal to 0. In that case the integral reduces to
dx exp − xt
Ax
=
2
(2π)n/2
. (B.9)
det(A)
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