Page 309 - Determinants and Their Applications in Mathematical Physics

294 6. Applications of Determinants in Mathematical Physics

Hence, the application of transformation γ to P n gives P .

n
In order to prove that the application of transformation β to P gives

n
P n+1 , it is required to prove that
ρ
φ n+1 = ,
φ
n
which is obviously satisfied, and
ωρ ∂ψ
∂ψ n+1
= − n
∂ρ (φ ) 2 ∂z

n
∂ψ n+1 ωρ ∂ψ
= n , (6.10.19)
∂z (φ ) 2 ∂ρ

n
that is,
2
n−2
n+1
n−1
∂ (−1) ωρ ρ ∂ (−1) ωA 1n

n
= −ωρ ,
∂ρ E n1 A 11 ∂z ρ n−2
2
n−2
n−1
n+1

∂ (−1) ωρ ρ ∂ (−1) ωA 1n

n
= ωρ
∂z E n1 A 11 ∂ρ ρ n−2
2
(ω = −1). (6.10.20)
But when the derivatives of the quotients are expanded, these two relations
are found to be identical with the two identities in Lemma 6.10.4 which
have already been proved. Hence, the application of transformation β to
P gives P n+1 and the theorem is proved.

n
The solutions of (6.10.1) and (6.10.2) can now be expressed in terms of
the determinant B and its cofactors. Referring to Lemmas 6.17 and 6.18,
ρ n−2 B n−1
φ n = ,
B n−2
n n−2
(−ω) ρ 2
ψ n = − B 1n (ω = −1),n ≥ 3, (6.10.21)
B n−2
B n−1

φ = ,
ρ B n
n n−2
n

ψ = (−ω) B 1n , n ≥ 2. (6.10.22)
ρ B n
n n−2
The first few pairs of solutions are

ρ −ωρ

P (φ, ψ)= , ,
1
u 0 u 0
P 2 (φ, ψ)=(u 0 , −u 1 ),

u 0 u 1
P (φ, ψ)= 2 2 , 2 2 ,

2
u + u 1 u + u 1
0
0
2 2 2
ρ(u + u ) ωρ(u 0 u 2 − u )
1
1
0
P 3 (φ, ψ)= , . (6.10.23)
u 0 u 0