Page 350 - Determinants and Their Applications in Mathematical Physics

A.11 Solutions of a Pair of Coupled Equations 335

Proof of (a). Put x =sh θ. Then,
1
g n = (e 2nθ + e −2nθ )
2
=ch 2nθ,
g m+n + g m−n = ch(2m +2n)θ + ch(2m − 2n)θ
=2 ch 2mθ ch 2nθ
=2g m g n .
The other identities can be verified in a similar manner.
It will be observed that

g n (x)= i T n (ix),
2n
where T n (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,

∂u r+1 ∂u r ru r+1
+ = − , r =0, 1, 2,..., (A.11.1)
∂ρ ∂z ρ
∂u r−1 ∂u r ru r−1
− = , r =1, 2, 3,..., (A.11.2)
∂ρ ∂z ρ
can be obtained in the form of a contour integral by applying the theory
of the Laurent series. The solution is
2
2
ρ 1−r , ρ w − 2zw − 1
dw
u r = f , (A.11.3)
2πi w w 1+r
C
where C is a contour embracing the origin in the w-plane and f(v)isan
arbitrary function of v.
The particular solution corresponding to f(v)= v −1 is
ρ 1−r , dw
u r =
2
2
2πi w (ρ w − 2zw − 1)
r
C
ρ −1−r , dw
= , (A.11.4)
2πi w (w − α)(w − β)
r
C
where
+
1 - 2 2 .
α = z + ρ + z ,
ρ 2
+
1 - 2 2 .
β = z − ρ + z . (A.11.5)
ρ 2