82 BELL SYSTEM TECHNICAL JOURNAL 



Splitting the interval of integration (— <», w^) into (— cc, — log ^X) and 

 (— log A\, icq) in the first integral on the right of (5-47) shows that its con- 

 tribution to /(X) is 



dy d%ce-''''-^'''''^ A / dy \ dw e~''''-^""' (5-48) 



00 •'—00 «'— oo J— log A\ 



Integrating with respect to y, after inverting the order of integration, shows 

 that the value of the first integral is 



tT'" j r" dl = {1+ erf B)/2 (5-49) 



where, from (5-37) and the definition (5-22) of Di, 

 B = -Kl + ry'\-^l'~ log^X 



1/2 -1/2 ,_ XA>(1 -f l/rT" (5-50) 



= -i(l+.)-.-^'Mog 



[27r9(l -f 2r)]i/2 



That the value of /(X) differs from (5-49) by 0(5"^''') may be seen as 

 follows. Since < exp \—^"/2q] < 1, the integral over (wo, °^) (mentioned 

 just above (5-46) and obtained by taking the limits of integration to be Wo 

 and <=o in the left side of (5-47)) is positive and less than 



/ expl-e'-'lrfw = / e-'dx/x = .219... (5-51) 



Likewise, the second integral on the right side of (5-47) is less than 



["" (1 - exp [- e"-"'°J) dw = f (1 - e-') dx/x = .796... (5-52) 



Therefore the contribution of the first integral on the right of (5-47) differs 

 from /(X) by a quantity less than 



[ A ^"'"'(.219 -1- .796) dy = 0(^"''') 

 ''—00 



in absolute value. The contribution of the first integral on the right of (5-47) 

 differs from (5-49) by the second integral in (5-48) which is 0(^~^''-) because 

 it is less than 



r Ih{yD)e-'''' dy 



•'-co 



The factor {y'-D) arises from 7<:'o — (— log .IX) when the mean value theorem 

 is applied to the integral in iv. Hence /(X) differs from (5-49) by 0(^"'''-). 

 Although (5-49) is a sufiicicntly accurate expression of ./(X) for our pur- 



