92 BELL SYSTEM TECHNICAL JOURNAL 



when q > \, 



b-'+c^ q^-\> {q-iy 



\v,\< \/{2q - 1) (A2-10) 



V2 < -2q+ \ 



The curve obtained by plotting vi as a function of y plays an important 

 role because, as we shall show, the maxima and minima of the curve for v 

 lie on it. Therefore, the maximum value of | i) | cannot exceed the maximum 

 value of \vi\. The maxima and minima must lie on either the Vi or the v^ 

 curve since v' vanishes only on these curves. In order to show that it is the 

 Vi curve we note from (A2-9) that, near y = ^, Vi behaves like y/{2q -\- 1). 

 Consequently both the Vi and v curves start from i' = at ;y = but for a 

 while vi lies above v which behaves like y/{2q -\- 2). Here v lies in a v' > 

 region and continues to increase until it intersects vi (as it must do before 

 y reaches 2 because v\ = at y = 2) at which point v' = 0, Vi ^ 0, and v 

 has a maximum which is less than the maximum of | Vi | so 7' < \/{2q — 1) 

 when q> \. Upon passing through vi, v enters a v' < region and decreases 

 steadily until it either again intersects the Vi, curve or else approaches some 

 limit as y ^ 00. In either case | v | does not exceed l/(2q — 1), since, in the 

 first case v would have a minimum at the intersection and in the second 

 Vi — >^ as y ^ 00. The same reasoning may be applied to the remaining 

 points of intersection, if any, of the v and vi curves. 



In order to obtain an inequality for V itself we rewrite (A2-6) as 



v' = c - {2b + v)v (A2-11) 



The solution of this equation which behaves like y/i2q + 2) as y -^ also 

 satisfies the relation 



v{y) = f c{x) exp - [ [2b(0 + vm d^ dx. 



Jo {_ Jx 



as may be verified by making use of the relations r(.v) -^ 1 as .v — > and 

 2b{0 -^ (2q + 1)/^, t(0 -^ ^/{2q + 2) as ^ -> 0. For then 



- [ [2b(^) + v(0] d^ -^ (2q + 1) log x/y 



Jx 



v(y) -> r (x/yY"'-' dx = y/{2q + 2) 

 Hence, from (A2-8) 



Viyi) = f dy £ dx) t-^P [-£ 12M^) + vm dn dx 



