Page 307 - Determinants and Their Applications in Mathematical Physics

292 6. Applications of Determinants in Mathematical Physics

which is equivalent to (b). This completes the proof of Lemma 6.20.
Exercise. Prove that
(n+1) (n+1)
∂ p − q − 1 A n+1,q ∂A pn ∂ A p+1,1 ∂A

p,q−1
ω − A pq = − n + A 1q + n ,
∂ρ ρ n A n ∂z n ∂z A n ∂z
(n+1)

∂A pq A ∂ n ∂ 1 A p+1,1

(n+1)
ω n = n+1,q − A pq − A 1q −
∂z A n ∂ρ ρ n n ∂ρ ρ A n
∂ q − 1 p,q−1 p +1

− − A − A p+1,q
∂ρ ρ n ρ n
2
(ω = −1).
Note that some cofactors are scaled but others are unscaled. Hence, prove
that

∂ n − 2 E n−1 E n ∂ A n−1 A n−1 ∂
ω − = − E n ,
∂ρ ρ A n A n ∂z A n A n ∂z A n

∂ E n−1 ∂ n E n−1
ω =(−1) n E n −
∂z A n A n ∂ρ ρ A n

A n−1 ∂ 1
+ − E n .
∂ρ ρ A n
A n
6.10.3 The Intermediate Solutions
The solutions given in this section are not physically significant and are
called intermediate solutions. However, they are used as a starting point in
Section 6.10.5 to obtain physically significant solutions.
Theorem. Equations (6.10.1) and (6.10.2) are satisfied by the function

pairs P n (φ n ,ψ n ) and P (φ ,ψ ), where

n n n
ρ n−2 A n−1 ρ n−2
a. φ n = = 11 ,
A n−2 A n−1
ωρ n−2 E n−1 (−1) ωρ n−2 (−1) n−1 ωρ n−2
n
b. ψ n = = n−1,1 = A 1n ,
A n−2 E A n−2
n−1
A 11

c. φ = ,
ρ
n n−2
(−1) ωA 1n 2
n

d. ψ = (ω = −1).
ρ
n n−2
The first two formulas are equivalent to the pair P n+1 (φ n+1 ,ψ n+1 ), where
ρ n−1
e. φ n+1 = ,
A 11
(−1) n+1 ωρ n−1
f. ψ n+1 = .
E n1