462 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 



S,{h) = Sn + S22 + So, 

 Sn = - [ F (In, 



J B 



S,2= -[ i^f',.(^o) dn/Wr.U), (7.16) 



&3 = / FVniz'o) dn/WnU), 



Jc 



F = e'" sec {d/2){iw/2yr{n + l)l\{z)Wn{z')/2i sin ttw. 

 When instead of (5.4) the expression (7.5) for S^ih) is used we obtain 



S21 '^ —Ml I i'^jS" dn/s'm irn, 



J B 



S,, Ml / i"'^"UnU) dn/[sm irnWnU)], (7.17) 



J A 



S2Z -- Ml f r"/3'T„(2o) dn/[sm TnWnU)]. 

 Jc 



In S22 it is permissible to swing AC from its original position BC 

 because the zeros of Un(Z(^) cancel those of sin irn. When /3 = 1, AC and 

 CD are the paths of steepest descent for AS22 and *S23 in (7.17) because 

 Fm [m - f{to)] = on ACD. 



The asymptotic expression (7.17) for Sn may be evaluated bj^ tem- 

 porarily assuming jS to be a complex number with \ \ < 1 and | arg \ 

 < 'w/2. The integral along BC is the integral along BCE minus the in- 

 tegral along CE (see Fig. 6.2). Deforming BCE into Li of Fig. 5.1 shows 

 that its contribution to S21 is —2iMi/(i — (8). An infinite series for the 

 integral along CE may be obtained by expanding i "/sin irn in powers 

 of exp ( — i2irn) and integrating term wise from n = no = — I — ih to 

 n = 'x> —ih, i.e., from C to E. In this way we obtain 



S21 2iMi 



+ E 



exp (no log (8 — 2Tth) 



(7.18) 



t=o log ^ — 2x?7 



Despite the appearance of the right hand side, it is analytic around 

 /3 = 1 and analytic continuation may be used to show that (7.18) 

 holds f or < /3 < -^ . 



When h is large only the first term in the series is important and 

 we have 



S21 - 2iMi[(^ - 1)"' - fS-'-"'Aog ^] 



(7.19) 

 = 2iMii^ - ir' -f iM2g~' 



