Page 301 - Determinants and Their Applications in Mathematical Physics

286 6. Applications of Determinants in Mathematical Physics

Hence, the first term of F is given by
∗
A x A =(−1) n+1 Q n+1,n+2 Q n+2,n+1 . (6.9.28)
x
Differentiating (6.9.26) and referring to (6.9.6),

A xx = ω ∂A rr
∂x
c r
r

= ω ∂A ss ∂θ s
∂θ s ∂x
c r
r s

= − c r c s A rs,rs
r s

= † c r c s A rs , (6.9.29)
r s

A t = ∂A ∂θ r
∂θ r ∂t
r
2
= −ω c A rr . (6.9.30)
r
r
Hence, applying (6.9.13) and (6.9.19),
2
A xx + ωA t = † c r c s A rs +
c A rr
r
r s r

= c r c s A rs
r s
= P n+2,n+2
= Q n+2,n+2 + Q. (6.9.31)
Hence, the second term of F is given by
∗
A (A xx + ωA t )=(−1) (B − Q n+1,n+1 )(Q n+2,n+2 + Q). (6.9.32)
n
Taking the complex conjugate of (6.9.31) and applying (6.9.20) and
(6.9.15),
∗
∗
(A xx + ωA t ) = P n+2,n+2
=(−1) n+1 (Q n+2,n+2 − Q). (6.9.33)
Hence, the third term of F is given by
∗
A(A xx + ωA t ) =(−1) n+1 (B + Q n+1,n+1 )(Q n+2,n+2 − Q). (6.9.34)
Referring to (6.9.14),
1 (−1) A (A xx + ωA t )+ A(A xx + ωA t ) ∗

∗
n
2
= BQ − Q n+1,n+1 Q n+2,n+2
= QQ n+1,n+2;n+1,n+2 − Q n+1,n+1 Q n+2,n+2 .