Page 351 - Determinants and Their Applications in Mathematical Physics

336 Appendix

This solution can be particularized still further using Cauchy’s theorem.
First, allow C to embrace α but not β and then allow C to embrace β but
not α. This yields the solutions

ρ −1−r −ρ −1−r
, ,
β (α − β) α (α − β)
r
r
but since the coupled equations are linear, the difference between these two
solutions is also a solution. This solution is
ρ −1−r (α + β ) (−1) f r (z/ρ)
r
r
r
, (A.11.6)
(αβ) (α − β) 1+ z /ρ 2
= +
2
r
where
+ + .
1
-
f n (x)= (x + 1+ x ) +(x − 1+ x ) . (A.11.7)
2 n
2 n
2
Since z does not appear in the coupled equations except as a differential
operator, another particular solution is obtained by replacing z by z + c j ,
where c j is an arbitrary constant. Denote this solution by u rj :
(−1) f r (x j )
r
u rj = 3 , x j = z + c j . (A.11.8)
1+ x 2 ρ
j
Finally, a linear combination of these solutions, namely
2n

u r = e j u rj , (A.11.9)
j=1
where the e j are arbitrary constants, can be taken as a more general series
solution of the coupled equations.
A highly specialized series solution of (A.11.1) and (A.11.2) can be ob-
tained by replacing r by (r−1) in (A.11.1) and then eliminating u r−1 using
(A.11.2). The result is the equation

2 2 2
∂ u r
− 1 ∂u r − (r − 1)u r + ∂ u r =0, (A.11.10)
∂ρ 2 ρ ∂ρ ρ 2 ∂z 2
which is satisfied by the function

u r = ρ {a n J r (nρ)+ b n Y r (nρ)}e ±nz , (A.11.11)
n
where J r and Y r are Bessel functions of order r and the coefficients a n and
b n are arbitrary. This solution is not applied in the text.