Page 291 - Determinants and Their Applications in Mathematical Physics

276 6. Applications of Determinants in Mathematical Physics

w xxxx =2V n−3,n,n+1 +3V n−3,n−1,n+2 +3V n−2,n−1,n+1 + V n−3,n−2,n+3 ,
w z = −V n−3,n−1,n + V n−3,n−2,n+1 ,
w zz =2V n−3,n,n+1 − V n−3,n−1,n+2 − V n−2,n−1,n+1 ,
w t = −4(V n−2,n−1,n − V n−3,n−1,n+1 + V n−3,n−2,n+2 ),
w xt =4(V n−3,n,n+1 − V n−3,n−2,n+3 ). (6.7.64)
Each of the sections in the second line of (6.7.60) simplifies as follows:
w t +4w xxx =12V n−3,n−1,n+1 ,
(w t +4w xxx ) x = 12(V n−2,n−1,n+1 + V n−3,n,n+1 + V n−3,n−1,n+2 ),
w xxxx − w zz =4(V n−2,n−1,n+1 + V n−3,n−1,n+2 ),
(w t +4w xxx ) x − 3(w xxxx − w zz )=12V n−3,n,n+1
2
w 2 − w =4V n−3,n−1,n V n−3,n−2,n+1 . (6.7.65)
xx z
Hence,
1
F = V n−3,n−2,n−1 V n−3,n,n+1 + V n−3,n−2,n V n−3,n−1,n+1
12
+V n−3,n−1,n V n−3,n−2,n+1 . (6.7.66)
Let

,
T
C n+1 = α 1 α 2 ...α n

,
T
C n+2 = β 1 β 2 ...β n
where
α r = D (ψ r )
n
x
β r = D n+1 (ψ r ).
x
Then
V n−3,n−2,n−1 = A n ,
(n)
V n−3,n−2,n = α r A ,
rn
r
(n)
V n−3,n−1,n+1 = − β s A r,n−1 ,
s
(n)
V n−3,n−2,n+1 = β s A ,
sn
s
(n)
V n−3,n−1,n = − α r A ,
r,n−1
r
(n)
V n−3,n,n+1 = α r β s A . (6.7.67)
rs;n−1,n
r s
Hence, applying the Jacobi identity,
1 (n) (n) (n)
α r β s A + α r A β s A s,n−1
12 F = A n rs;n−1,n rn
r s r s