Determinants and Their Applications in Mathematical Physics

256 6. Applications of Determinants in Mathematical Physics

=

ρ

2

Bn+1Bn− 1 e

− 2 x

B

2

n

=

Bn+1Bn− 1

B

2

n

This equation is identical in form to the equation in the corollary to

Theorem 6.3. Hence,

Bn=

∣

∣Di+j−^2

x

g(x)

∣

∣

n

,g(x) arbitrary,

which is equivalent to the stated result.

Theorem 6.5. The equation

(D

2

x

+D

2

y

) logun=

un+1un− 1

u

2

n

is satisfied by the function

un=An=

∣

∣Di−^1

z

D

j− 1

̄z (f)

∣

∣

n

,

wherez=

1

2

(x+iy),z ̄is the complex conjugate ofzand the function

f=f(z, ̄z)is arbitrary.

Proof.

D

2

x(logAn)=

1

4

(

D

2

z+2DzD ̄z+D

2

̄z

)

logAn,

D

2

y

(logAn)=−

1

4

(

D

2

z

− 2 DzDz ̄+D

2

z ̄

)

logAn.

Hence, the equation is transformed into

DzD ̄z(logAn)=

An+1An− 1

A

2

n

,

which is identical in form to the equation in Theorem 6.3. The present

theorem follows.

6.5.3 The Milne-Thomson Equation............

Theorem 6.6. The equation

y

′

n

(yn+1−yn− 1 )=n+1

is satisfied by the function defined separately for odd and even values ofn

as follows:

y 2 n− 1 =

B

(n)

11

Bn

=B

11

n

,

y 2 n=

An+1

A

(n+1)

11

=

1

A

11

n+1

,