Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 116

This system includes the solution of the previous auxiliary system (2.112) as well
as the definition

β 1 = a 1,0 r 2 + a 1,1 r 1 . (2.118)

Reversing the order of equations and variables, and taking complex conjugates,
leads to the system

0, a ∗ 1,1 , a ∗ 1,0 R 2 β ∗ 1 ,0,α ∗ 1 .

= (2.119)

Multiplying this system by k 1 and adding the result to the original system (2.117)
gives

a 1,0 , a 1,1 +

k 1 a ∗ 1,1 , k 1 a ∗ 1,0 R 2 =

α 1 +

k 1 α ∗ 1 . k 1 β ∗ 1 , 0,β 1 + (2.120)

For this to solve the next auxiliary system,
a 2,0 , a 2,1 , a 2,2 R 2

= (α 2 ,0,0), (2.121)

requires

=− β 1 /α ∗ 1 , k 1

a 2,0 = a 1,0 ,

k 1 a ∗ 1,1 , a 2,1 = a 1,1 +

(2.122)

k 1 a ∗ 1,0 ,
α 2 = α 1 +

a 2,2 =

k 1 β ∗ 1 .

= +

The solution of the auxiliary system for m n 1 is then derived from the solution

for m = n through the recurrence relations

β n = a n ,0 r n + 1 + a n ,1 r n +···+ a n , n r 1 ,

=− β n /α ∗ n , k n

a n + 1,0 = a n ,0 ,

k n a ∗ n , n , a n + 1,1 = a n ,1 +

. . .

(2.123)

k n a ∗ n ,1 , a n + 1, n = a n , n +

k n a ∗ n ,0 ,
α n + 1 = α n +

a n + 1, n + 1 =

k n β ∗ n .
A similar scheme can be devised to solve the full equations for m = 1. We begin

with the system

⎝ ⎠ =

r 0 r 1

f 0,0 ,0

(g 0 ,γ 0 ), (2.124)

r ∗ 1 r 0