Kh2
=K4 +m +m(g g) -1 ,m,, (C-7)
4+ 2
S(23,,0o + m,2 + m,-2)B +(q f +r a 8 I
-K (36,, + 3 -1 +m,3 + ,-3), (C-8)
96. .
and
-(3-,+3, )B-m n ) r)+ (q ))+f [(m+)(q1 q~ -q 1)+(m -)(qq -qq)]
Ka(26,o + m,+,2 + m2 ()+m(g gi) m, (C-9)
16z 2-
From Equation 4-43 and Equation 4-34, we obtain
exp(nh)ri E,,, 0 (C-10)
m= c
and
exp(nh)q'- q M =E, = 0. (C-11)
m= co
Those six equations (Equations C-6 to C- 1) can be further simplified by eliminating {r)
and ({q~ from equation set. We take index n as a positive integer only in the analysis.
Rewrite Equation C-ll as
1 +C
qn = -exp(-nh) m mln mq (C-12)
m=- co
and take the negative indices,
1 +C
q =-exp(nh) mI+ mq (C-13)
YII m= -co
where the (n-m)th order modified Bessel function of first kind for argument -na, In m (-na), has
been written as In-m for simplification purpose. Similarly, In+, denotes the (n+m)th order
modified Bessel function of first kind for argument na, In+, (na). The same format is used from
this point forward in this appendix. Equation C-10 is rewritten as
1 +
r= -exp(-nh) M mI- ,,m (C-14)
Sm= -1co
and
1 +CO
r =-exp(nh) mi mr (C-15)
YII m= -co
Change the dummy variable in Equation C-6 and Equation C-7 from m to n, and use the
expressions in Equations C-12 to C-15 to eliminate i r~, q' and q~, then we obtain
Sm -I+ ri + V mh nm In+q 0] (C-16)
=o exp(2nh) n+m ,, exp(2nh)0 (