118 BELL SYSTEM TECHNICAL JOUkNAL 



The results corresponding to Rj, are quite similar. WTien the corner 

 is not truncated 



r,(2) . 2 v^ T(n - ic)T(n + id v^ (- 2/3) 2m (i3),i_m 



^w = —''K 2^ , i — T-ST/^ TVTTT 2^ 



;f='i in + !)!(« - l)!2c^o (2w)!(« - w) 

 2c 



• 2 



y - ^'(1 - iO - ^3 (2 + ^')(l - i2) (6-6) 



- 2^(23 + 20/3^ + 2/3')(l - ig) - 



i - 1 - ^(1 - /S) - .5772 



(1/2 - ^),. 



- ^(1 - /3) E 



^i(l/2)„m(m - ,8)(m - ^+ 1) 



in which ^„ is obtained by replacing c by ^ in the expression (6-4) for A^ . 

 The evaluation of the integrals for i?^ and R^ for general values of / 

 appears to be difficult although it is possible to obtain approximate expres- 

 sions for the case when t is large. 



7. Reflection from Small Angle Corners 



The expressions for i?^^^ and R^^^ may be evaluated approximately when 

 the angle of the corner is small. It turns out that, for / = 0, they are of the 

 same order of magnitude and both of them must be considered. Moreover 

 i?<"' for n> 3 differs from R^^^ by terms of the same order as those neglected 

 in our approximations so that there is no point in going to the higher values 

 of n. 



We first obtain the approximation for Re for a corner with no truncation 

 having the total angle t^. Since ,8 is very small (6-1) may be written as 



g{v, &) - exp \^<p] - 1 = /3v^ + /3VV2: + 0(/5«) 



(7-1) 

 if ^ log {chv -\- cos 6) — log {chv — cos 6) 



where 0(i3') denotes terms of order jS^ The expression ^p becomes very large 

 near the two points (0, 0) and (0, tt) (the coordinates being {v, 6)). The 

 following considerations indicate that this does not in\-alidate our procedure. 

 The remainder, denoted by 0((8'), in (7-1) is less than \^ip\^ exp [ jS^ |. 

 Near (0, 0) ip is approximately equal to 21og(2/r) where r- = v^ -\- 6^. 

 Consequently the remainder is less than {2j3 log 2/r)* {2/rY^. \A'hen the 

 expression (7-1) for g(v, 6) is set in the integral equation it is seen that all 

 terms, and in particular the remainder term (by virtue of the inequality 

 just stated), of the double integral converge at (0, 0). Hence the contribu- 

 tion of the remainder term is of order ^ even in the worst case when the 



