370 BELL SYSTEM TECHNICAL JOURNAL 



Let te — u , u = e 

 € — i2c\(T = a = 1/t^ 

 e-(h+znVat cos y^ Vat = §(e"*'' + e~'"') 

 g' = (h -\- z — iy)^ \/ a 

 f={}i-\-z^iy)lVa 

 g = {h + 2)r Va 

 Since it' + uY" = {f + 2tu + u' - Ituf" 



= t + u - tu/(t + u) + 



we put 



l/W/' + w2 + at) = 1/it -\- u - tu/{t + w) + at) 

 = (/ + '«-)/[(« + 1)/' + (« + l)/w + u^ 

 = (/ + n + r2)/ (a+ l){t + r,) {t + r^) 

 where fi = u{l - \/ l - 4/(a + l)) /2. 

 and r2 = w(l + Vl - 4/(a + l))/2. 

 Then 



(a + l)(r2 — ri) Jo V + n / + ^2/ 



2 

 1 



+ ' -^dt 



[-r,e' '■^ li {e-" "') + rje"^ '' li (e"^ '■^) 



2(a + l)(r2 - n) 



- ^26^"^^ li (e-^"'0 + ne^"'^ li {e~'"'')]- (9) 

 When )> = this reduces to 



M + W ^ 7— r-T^ S [-^2^"^' li (e-^'O + ne"'"' li (e-''^')]. (10) 



(a 4- l)(r2 - ri) 



^i (^~') = - f ^dt = C + ]ogz + J^ 



Jz t t=l 



(-2)' 



t=l tit 



where C = .577215665 and, ii z = re" , —x<d<Tr. 



-2 n 



— z t«o (— z;* 



The accompanying charts give M and N ior y = and e = 15. With 

 z = h they give M and A^^ for />ii . The computed points are indicated by 

 solid dots on the chart for M\ they were obtained by numerical integration. 



The approximation (10) was checked against the values obtained by 

 numerical integration at a number of points. The discrepancy in each case 

 amounted to less than one per cent for both M and N. This approximation 

 is a much easier way to evaluate the integral than is numerical integration 

 but it is a tedious computation with many chances for error. Conse- 



