Rewriting the P piece and using Equation C-11 with p = 1 leads to
19(pim) + -r (1 + poft Cl = 0, (C-17)
p p
which, after substituting Equation C-3, the complex conjugate of Equation C-4 and
Equation C-15 along with some re I1 llpil_ yields
Integration then gives us
m mol~ _I 0 lo a lo _~ ) lo _C19)
p p
and the solution for (m then follows from complex conjugation
m7 = (mo ~o a_ o.(-0
p p
Finally, we are in a position to deal with (n, by writing
P'e + Des + (-r + -r')(m + (-r + -r')(m = 0, (C-21)
in terms of Held's operators (Equations C-1, C-3 and C-4) as
en>(I t~rt-I f- -6
P~+ ~-rllp p1)E
(C-22)
+4P ), ~+ (7 + v'r)(m + (7 + -y')(m = 0.
Substituting Equations C-3, C-4, C-5, C-15, C-19 and C-20, rearranging terms and
letting the dust settle leads to
~;=-p~ 1 1 -a o 1 1\
2 2 pp
,I 1 1, (C- 23)
[xro 8 + cto a or( a +l~ 2xo~mo + 2 o mo
p p
2 2