80 BELL SYSTEM TECHNICAL JOURNAL 



Combining these and a similar expression for log vi leads to 



mF(vi) — — m log (1 + 1/r) + 7 — 5 



-[(1 + r)y - r5]V[2w(l + 2r)] + 0(w-'/2 iog3/2 ^„) 



(5-34) 

 = -{q + 1) log (1 + \/r) + a - /3 - [(1 + r)a - r^fD 



+ Q{q-'i-' log3/2 q) 



Substitution of (5-30) and (5-34) in (5-27) gives the result we seek: 



P(xo, w) = (1 + l/rY'^-KlT^qyi'D, 



(5-35) 

 exp (a - ^ - [(1 + r)a - r&\-D + 0(g-i/-' log^^^ ^)) 



Since P(.Vo, «) occurs only in the product KP{xo, u) in (5-26) we set, in 

 view of (5-35), 



KP{x% u) = A\(oc, (3) exp S(a, (3) (5-36) 



where \(a, /3) stands for the terms denoted by exp [0(^~''- log^'- q)] in (5-35) 

 and 



A = A'(l + lA)-«-i(27r^)i/-/)i 



(5-37) 

 S{a, 13) = a- ^-[{l + r)a- r/3]-/> 



As long as I a I < i and | /3 | < f,\{a, (3) is nearly unity and we write 



Xi < X(a, /3) < X2 



(5-38) 

 Xi = 1 - e, X2 - 1 + €, e = Cq-"'- log^'^ ^ 



where C is a positive constant large enough to make e dominate the terms 

 of order q~^''' log^'- q in (5-35). q is supposed to be so large that e is very small 

 in comparison with unity. 

 Setting (5-36) in (5-26) gives 



Prob. (PiQ, ■■■ , PkQ > PoQ) = / + 0(1/A-) -f 0(r'/- Iog'^2 q) (5-39) 



where the contribution of the region outside | a | < /, | /S | < I has been 

 returned to the terms denoted by 0(^~''"' log^'- q) (we could have stayed in 

 the region | a | < f,\f3\ < ( from (5-23) onward, but didn't do so because 

 we wanted to show that the results coming from (5-25) were not restricted 

 to this region) and 



( ( 



I = j da j (//3 Di exp [- Q{a, /3) - ^X(a, ^)e'^"'''^] (5-40) 



Let L(X) denote the integral obtained by replacing the function X(a, /3) in 

 / by the [)ositive constant X (which we shall take to be either Xi or X2 dehned 



