Determinants and Their Applications in Mathematical Physics

294 6. Applications of Determinants in Mathematical Physics

Hence, the application of transformationγtoPngivesP
′
n

.

In order to prove that the application of transformationβtoP

′

n

gives

Pn+1, it is required to prove that

φn+1=

ρ

φ′

n

,

which is obviously satisfied, and

∂ψn+1

∂ρ

=−

ωρ

(φ′

n

)^2

∂ψ

′

n

∂z

∂ψn+1

∂z

=

ωρ

(φ

′

n)

2

∂ψ

′

n

∂ρ

, (6.10.19)

that is,

∂

∂ρ

[

(−1)

n+1

ωρ

n− 1

E

n 1

]

=−ωρ

[

ρ

n− 2

A

11

] 2

∂

∂z

[

(−1)

n

ωA

1 n

ρ

n− 2

]

,

∂

∂z

[

(−1)

n+1

ωρ

n− 1

E

n 1

]

=ωρ

[

ρ

n− 2

A

11

] 2

∂

∂ρ

[

(−1)

n

ωA

1 n

ρ

n− 2

]

(ω

2

=−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 transformationβto

P

′

ngivesPn+1and the theorem is proved. 

The solutions of (6.10.1) and (6.10.2) can now be expressed in terms of

the determinantBand its cofactors. Referring to Lemmas 6.17 and 6.18,

φn=

ρ

n− 2

Bn− 1

Bn− 2

,

ψn=−

(−ω)

n

ρ

n− 2

B 1 n

Bn− 2

(ω

2

=−1),n≥ 3 , (6.10.21)

φ

′

n

=

Bn− 1

ρ

n− 2

Bn

,

ψ

′

n

=

(−ω)

n

B 1 n

ρ

n− 2

Bn

,n≥ 2. (6.10.22)

The first few pairs of solutions are

P

′

1 (φ, ψ)=

(

ρ

u 0

,

−ωρ

u 0

)

,

P 2 (φ, ψ)=(u 0 ,−u 1 ),

P

′

2

(φ, ψ)=

(

u 0

u

2

0

+u

2

1

,

u 1

u

2

0

+u

2

1

)

,

P 3 (φ, ψ)=

(

ρ(u

2

0 +u

2

1 )

u 0

,

ωρ(u 0 u 2 −u

2

1 )

u 0

)

. (6.10.23)