122 BELL SYSTEM TECHNICAL JOURNAL 



The Fourier coefficients of g(z), 6) may be obtained by using the results of 

 Appendix II. Assuming z) > 0, w > 0, 



a^iy) = 2a{v - t^v) + 2-'a^^h{v, v) + hiv - t, v - l) 



+ Ii(v -\- t,v-\-t) - -ihiv - t,v) - 4/2(7) -}- t,v) + 2 h{v - l,v-^ t)], 



amiv) = 2am-'[-2(-)"'e-^\''\ + g-^l'^'l + g-^l^+'l] (7-13) 



where xpiv) = 1 when < v < t and i^(d) = when v > t. Substitution 

 of the values (A2-3) for /i and h gives 



ao(i') = [2a{v - + 2a2(^ - 0']^(i') (7-14) 



n=l 



-f g-2nlt^-(| ^ 2g-''l''-«l-"l"+'l — 4(— )"e-"I'^'l^''] 



The second approximation to the reflection coefficient is 

 R'i^ = iak-' sh-i'kt - ia^k-n-'{2kt - sin 2kt) 



- ika'^T. n-'in' + k')'' {2-(-)"2e-»' (7-15) 



+ [1 - 2(-)"e-"' + e-2"']cos 2kt 

 + wyfe-'[e-2»« _ (-)"2e-«']sin 2kt] 

 The typical term in the summation (4-5) ior Rg^ is 



- .— f ^" ^^0 a.(.o) r ^^' ^^(tO^"''''''^"'^"'''"'"''''" (M6) 



When w = 0, eo = 1, To = ik, and 00(1*) is 2a(i) - + 0(«") for < i' < / 

 and is 0(a^) for f > /. The integral may then be approximated by replacing 

 the upper limit 00 in (A2-14) by /. The value of (7-16) for m = is found 

 to be, to within 0(a^), 



2-ia¥(e-2'fc' - 1) - (3/4)to2^-*(sin 2kt - 2kt) (7-17) 



When m > 0, €„ = 2,^1, = m^ — k^, and the substitution of the value 

 (7-13) for amiv) enables us to express (7-16) as the sum of six /'s where J is 

 defined by (A2-4). The /'s may be evaluated with the help of (A2-7) and 

 (A2-8). Substitution of this value of (7-16) and the value (7-17) for m = 0, 

 together with R^e^ given by (7-15), in the expression (4-5) for R^ gives 

 our final result 



R^P = iak-h\n^ kt + a2/22-Kg-2"' - 1) (7-18) 



+ ia^[^-'k-\2kt - sin 2kt) - Bsm 2kt + ^Z >7--7^'^n] 



