342 BELL SYSTEM TECHNICAL JOURNAL 



and 



/. .= 7r-4e;2[w-i - {-)'v-^ - tt-^Js] (Al-8) 



except when 5 = in which case 



/o = Tr~^w\u~'^ — v~^) — 1 = w^/iuv) — 1 



(Al-9) 

 = iryiAuv) = [(2piA)2 - l]-i 



When pi/a is large, u and v are large, and the asymptotic expansion of 

 (Al-6) gives 



w~'Js -' s-'lu-' - (-)Vi] - 2ls-'[u~' - (-)V] + ••• (Al-10) 



When (Al-10) is placed in (Al-8) 



Is -- Tr~'2lwh~\u~^ - (-)V] - • . • (Al-11) 



Formulas for Ks may be obtained in much the same way. 



Ks - (pi/a) / cos s{t -u) dt/t 



J u 



(Al-12) 



= {pi/a)l[Ci{sv) — Ci{su)] cos su + [5i(5^) — 5z(^w)]sin su] 

 and when s = 



Ko = (pi/a) log (1 + ir/u) (Al-13) 



The asymptotic expression is 



apT Ks '^ 5~ [u~ — i—yv~ ] — 3ls~ \u~ — {—yv~ ] + • • • 



It is convenient to write the asymptotic expressions in terms of the new 

 variable 



^ = a/p, (Al-14) 



When s is even and greater than zero 



/. ^ e/s, Is '-^ 6^V-25-2, Ks ^ 2^V-2^-2 (Al-15) 

 and when s is odd 



/. ^ 2^/s, I, '^ 4^7^-25-^ Ks ^ 2^T-h-^ (Al-16) 



When 5 = 



/o -- ^/4, ifiTo -- 1 + ^V12 (Al-17) 



We shall need the following asymptotic expressions which may be obtained 

 from the above work 



