Page 271 - Determinants and Their Applications in Mathematical Physics

256 6. Applications of Determinants in Mathematical Physics
2
ρ B n+1 B n−1 e −2x
=
B 2
n
B n+1 B n−1
= .
B 2
n
This equation is identical in form to the equation in the corollary to
Theorem 6.3. Hence,
i+j−2
B n = D g(x) , g(x) arbitrary,

x
n
which is equivalent to the stated result.
Theorem 6.5. The equation
2
2
(D + D ) log u n = u n+1 u n−1
u
x y 2
n
is satisfied by the function
i−1 j−1
u n = A n = D D (f) ,

z ¯ z
n
where z = 1 (x + iy), ¯ is the complex conjugate of z and the function
z
2
z
f = f(z, ¯) is arbitrary.
Proof.
2
2
D (log A n )= 1 D +2D z D ¯z + D 2 log A n ,
4
x z ¯ z
2 1 2 2
D (log A n )= − D − 2D z D ¯z + D log A n .
4
y z ¯ z
Hence, the equation is transformed into
A n+1 A n−1
D z D ¯z (log A n )= ,
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 (y n+1 − y n−1 )= n +1
n
is satisfied by the function defined separately for odd and even values of n
as follows:
(n)
B 11 11
y 2n−1 = = B ,
n
B n
A n+1 1
y 2n = (n+1) = 11 ,
A 11 A n+1