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THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 



- 2m) < 7r/2)]: 



(a) Re [j{t\) — /(/o)] is almost zero between — i.^ and —ih. 



(b) Im [/(/i) — /(^o)] is almost zero between —ih and — f co. 



(c) tifta runs from zero to unity as n goes from — 1 to — l — ih. 



Items (a) and (b) are consistent with/(^o) — fih) ^ (/■izo){z'o^ — ^inf'' 

 which holds when n is near —ih and which was mentioned in connection 

 with (G.IO). 



This concludes our discussion of the reasons for believing that (7.5) 

 is true for general values of h. Now we shall check it for the special case 

 h = 0. 



When w^e set /i = (i.e. z'(, = 0) in (7.3), use (9.4) and close L2 by an 

 infinite semicircle, we obtain 



1 



^2(0) 





1 



2iTl + 



(7.6) 



This agrees with the first two terms in the asymptotic expansion of 

 the Fresnel integral expression (5.20) for ^§2(0). In expanding (5.20) we 

 need the first of the two asymptotic expansions (both of which hold 

 for T » 1) 



L 



-iti i exp (-?T-) 



e at '^ —- 



1 - 



/ 



-t<2 



(It 



W/iY" + 



2T 



iexp j-iT-) 

 2T 



and also the first of the relations 



;r sin 6 + y cos 1 



To = 77(1 + 0) cos (e/2). 



Tl 



■r, 



(7.7) 

 (7.8) 



(7.9) 



In much of the following work we shall assume ^ and 77 to be so great 

 that we can neglect the terms denoted by QQi/r^'') in (7.5). We shall use 

 the asymptotic sign '^ to acknowledge this omission. 



From (7.4), /3 is equal to unity when <p = B — x/2. When r is very 

 large this value of <p marks the shadow boundary. In the shadow /3 > 1 

 and in the illuminated region ^ < \. 



Closing L2 on the right and on the left converts (7.5) into 



SM ^ 2iMr Z ^"VniZo)/Wn{Zo), 



(7.10) 



