90 BELL SYSTEM TECHNICAL JOURNAL 



Using (Xx — ave. (x — xY = 4(m + m) = 4m (1 + 2/) it is found that 



vi= \ + e/(l + 2/) = 1 - 2(.To - x)al 



mFivi) = -me'/il + 4/) = -(.To - xY/lal 



Imbo = m(vi — 1)-(1 + 2/) = (xq — xf/di 



and that, since ii\ — > — <» , Jj — > 62 and 64 — > Ihi/Vl. When these values are 

 put in (Al-27) the leading term becomes 



P{x^, u) - (27r)-i/2((r./2) exp [-sV2(rI] 



and the term within the braces in (Al-27) reduces to 1 — allz where z = x 

 — xo > 0. Since the asymptotic expansion is useful only in the region where 

 the second term within the braces is small in comparison with the first term, 

 which is unity, x — .tq must be several times as large as Ox before we can use 

 (Al-27). It will be noticed that the above expression for P(to, w) is closely 

 related to the asymptotic expansion of the error function. 



APPENDIX II 



An Approximation for \ii{oc) 



When 2 in the Bessel function Jq{qz) is imaginary a formula given by 

 Meissel (12) becomes 



T (n'.,\ - (?y)' exp {qw -\r V) , . 



^'^^^^ - en\q + 1)^1/^(1 + ^a^y ^^^'^^ 



where w = (1 + y^)^!- and F is a function of y and q which, when q is large, 

 has the formal expansion 



^2,,,6 



24^ ( w^ ] 16q^w' 



1 f _ 16 -I- 1512/ - 3654/ -j- 375y \ 

 5760^3 \ w' j ^ 



(A2-2) 



Here we shall show that for y ^ and ? > 1 



\V\ < 1/(29 - 1) (A2-3) 



Consideration of (A2-2) and also of the method used to establish (A2-3) 

 indicates that the inequality is very rough. It doubtlessly can be greatly 

 improved (but not beyond the l/(l2q) obtained by letting y and 9 — > 00 in 

 (A2-2)). Incidentally, it may be shown that the constant terms which re- 

 main in (A2-2) when y = 00 are associated with the asymptotic expansion 

 of log T(q -f 1). 



