p(vy= =| zl p(v,0,z) dz 
0 
in} 
1 
S|] Fp@ ODE (12.161) 
N | p(v, 0,z) 
0 
Finally he derived the expressions for his first and second approximations pj(v), 
Pi(v) and p3(v), p2(v) as follows. First, by neglecting the term higher (than 3) order 
joint cumulants, the first approximation is, 
ih Dee v Sim v v(1—€? 1/2 
Pi) =e exp] Fa +(1-€7)'/? y exp es anGer ae 
(12.162) 
2 2\1/2 
Ty ee Vis 21/2 v WL=Ge)) 
PAW) = in exp oe —(1-¢7)!/2 y exp} -— en erae 
(12.163) 
where #(a) is the Gaussian cumulative distribution function and € is the spectrum band 
width parameter. 
The € was introduced in Eq. 12.162 and Eq. 12.163 from the relations as follows that 
comes from the characters of joint cumulants, 
Ko20 21/2 
Ane 3 —— = Sil 12.164 
101 & 2 discs) ( ) 
1 =A%5) = €? 6 (12.165) 
It is interesting to find that this first approximation, Eq. 12.162, is just the same as 
the one that Cartwright and Longuet—Higgins!> derived as shown in Eq. 12.49, and Fig. 
12.1 in Section 12.5. 
Se 
