and
nm [I-m +I, r- m -h In-m -+h q
m o exp(2nh) n m=- n h exp(2nh) n )h
(C
exp(nh)
Next, we multiply Inm(-na) on both sides of Equation C-8, and take summation from
m -co to m = +o It yields
2 +m +++
a [21n +In+ 2 + B+ [In-m +T r T +T + -+ 2 n + I.-1 m .-1 + n-m.1 qm
4 m=- m=-2
-17)
KcI [,-3 + 3,+ + 3;_, +31+1 (C-18)
96,7
A similar manipulation is applied to Equation C-9, we have
m+C +m
-a [I- I +I B- n- n m)# + n-m I4-a-i n-+ I n 4-i + 4 ) n
m=-m m=-w
K6 [2, + In2 + 1 + m[In m + Im]m 2 I (C-19)
16r m=-co 2 /
Taking the zeroth order of Equations C-7, C-8, C-9 and C-11, we obtain
Kh2 1
q0 + Bh =, (C-20)
8; 4;'
2
ro8__a(ql,8 + q8)+a-B = 0, (C-21)
2 4
q + -a I(q
Ka2 1
16;r 4;'
(C-22)
(C-23)
Sq a q -( -qPl).
*d ~l \4\ *-\2
Equations C-16 to C-19, supplemented by Equations C-20 to C-23, are sufficient to solve
the Fourier coefficient sets {(~ and {(q Once they are solved, q' and ir are determined
from Equation C-12 and Equation C-14; q4 and r are solved from Equation C-6 and Equation
C-7, as
qr = (2nh-1) q +2nr +n g(g g),
(C-24)
and
r = -h(q, + q,,)- i:. (C-25)
The zeroth term of {r~ and {(q} are obtained from Equation C-10 and Equation C-11 at
n = 0, namely, Equation C-23 and
r2 = r-(- rl). (C-26)
q