Determinants and Their Applications in Mathematical Physics

336 Appendix

This solution can be particularized still further using Cauchy’s theorem.

First, allowCto embraceαbut notβand then allowCto embraceβbut

notα. This yields the solutions

ρ

− 1 −r

β

r

(α−β)

,

−ρ

− 1 −r

α

r

(α−β)

,

but since the coupled equations are linear, the difference between these two

solutions is also a solution. This solution is

ρ

− 1 −r

(α

r

+β

r

)

(αβ)r(α−β)

=

(−1)

r

fr(z/ρ)

√

1+z

2

/ρ

2

, (A.11.6)

where

fn(x)=

1

2

{

(x+

√

1+x

2

)

n

+(x−

√

1+x

2

)

n

}

. (A.11.7)

Sincezdoes not appear in the coupled equations except as a differential

operator, another particular solution is obtained by replacingzbyz+cj,

wherecjis an arbitrary constant. Denote this solution byurj:

urj=

(−1)

r

fr(xj)

√

1+x

2

j

,xj=

z+cj

ρ

. (A.11.8)

Finally, a linear combination of these solutions, namely

ur=

2 n

∑

j=1

ejurj, (A.11.9)

where theejare 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 replacingrby (r−1) in (A.11.1) and then eliminatingur− 1 using

(A.11.2). The result is the equation

∂

2

ur

∂ρ

2

−

1

ρ

∂ur

∂ρ

−

(r

2

−1)ur

ρ

2

+

∂

2

ur

∂z

2

=0, (A.11.10)

which is satisfied by the function

ur=ρ

∑

n

{anJr(nρ)+bnYr(nρ)}e

±nz

, (A.11.11)

whereJrandYrare Bessel functions of orderrand the coefficientsanand

bnare arbitrary. This solution is not applied in the text.