COMMUNICATION IN PRESENCE OF NOISE 89 



is bounded except near w = Wi (i.e., r = 0) and, indeed, decreases to zero 

 like —e~^/s as w ^ oo ± t'tt (i.e., r ^ oo). 



The values of b-2, bs, hi obtained by expanding (Al-22) and comparing 

 the result with (A 1-25) are 



b, = [F"'(v,)e"''' + 3F"(v{)e'''']/6 (Al-28) 



b, = [F""(vy'' + 6F"'(vy"' + 7K(z'i)e'"'']/24 

 F"{v) = D-2 + 2/^-^ F"'(v) = -2v-' - 6/^-^ F"''(v) = 6v-' + 24/z)-5 



Our asymptotic expression for P(xo, u), when .vo < x, is given by (Al-28) 

 and (Al-27). Only the leading term of (Al-27) is used in the paper. Some- 

 times the following expressions are more convenient than the ones which 

 have already been given. 



b, = vT\v, + lOi'^'ll = v\\v, + 2/)(ri - 1)V2 



= (1 - lAi)2(l + 4s/)i/V2 (Al-29) 



F{v^ = (1 + 45/)i/2 _ ^ _ / _ log Vx. 



In all of these formulas v\ is given in terms of 5 and / by (Al-13) and s and 

 / in terms of Xo and u by (Al-7). 



When .To > X, the saddle point x\ lies between and 1 in the v plane. As 

 V follows the path of steepest descent (discussed just below equation (Al-18)) 

 arg {v — 1) now stays close to tt. From (Al-19) Imag. w stays close to x on 

 the new path of steepest descent in the w plane, and the saddle point W\ 

 now lies on the negative real portion of the line Imag. \v = tt. The new path 

 starts at w = =o + it, swings down a little as it comes in, swerves up to 

 pass through wi and then goes out to 2ei = °o + iir above the branch cut 

 joining w = iir iow = «^ + itt. The analysis goes along much as for V\ > 1 

 except that instead of being the imaginary part of Wi is jtt. This causes 

 the terms in bz and 64 containing exp {iw]) to change sign. The numerical 

 values of bo and F{vi) are computed by the formulas (Al-29) as before. The 

 fact that Z>2 contains the factor exp {H-k) shows up only in changing the sign 

 of h\ to give the minus sign in the leading term : 



P(.i-o, u) ^\ - (47rw| b-2 \)-'i~ exp [wF(i'i)] 



which holds for .Vq > x. The one arises from the pole at i) = 1 and is the 

 same as the one in the second of equations (Al-6). 



In order to see how (Al-27) breaks down near .vo = x, we set .to — x = 

 2m{s — 1 — /) = — 2me or s = 1 + / — € where e is a small positive number 



