Determinants and Their Applications in Mathematical Physics

A.11 Solutions of a Pair of Coupled Equations 335

Proof of (a).Putx=shθ. Then,

gn=

1

2

(e

2 nθ

+e

− 2 nθ

)

=ch 2nθ,

gm+n+gm−n= ch(2m+2n)θ+ ch(2m− 2 n)θ

=2 ch 2mθch 2nθ

=2gmgn.

The other identities can be verified in a similar manner.

It will be observed that

gn(x)=i

2 n

Tn(ix),

whereTn(x) is the Chebyshev polynomial of the first kind (Abramowitz

and Stegun), but this relation has not been applied in the text.

A.11 Solutions of a Pair of Coupled Equations

The general solution of the coupled equations which appear in Sec-

tion 6.10.2 on the Einstein and Ernst equations, namely,

∂ur+1

∂ρ

+

∂ur

∂z

=−

rur+1

ρ

,r=0, 1 , 2 ,..., (A.11.1)

∂ur− 1

∂ρ

−

∂ur

∂z

=

rur− 1

ρ

,r=1, 2 , 3 ,..., (A.11.2)

can be obtained in the form of a contour integral by applying the theory

of the Laurent series. The solution is

ur=

ρ

1 −r

2 πi

∫

C

f

(

ρ

2

w

2

− 2 zw− 1

w

)

dw

w

1+r

, (A.11.3)

whereCis a contour embracing the origin in thew-plane andf(v)isan

arbitrary function ofv.

The particular solution corresponding tof(v)=v

− 1

is

ur=

ρ

1 −r

2 πi

∫

C

dw

wr(ρ^2 w^2 − 2 zw−1)

=

ρ

− 1 −r

2 πi

∫

C

dw

w

r

(w−α)(w−β)

, (A.11.4)

where

α=

1

ρ

2

{

z+

√

ρ

2

+z

2

}

,

β=

1

ρ

2

{

z−

√

ρ

2

+z

2

}

. (A.11.5)