DIFFRACTION OF RADIO WAVES BY A I'AHAliOhIC CYMXDKR 449 



for »S2(/i): 



sec - 



i;c\" l\n -\- 1) 



2i Jl,\'^/ s 



sin TT/t 



Un{z)W„{z') 



rvr.izo) 



(o.ll) 



Vr^iz') 



.WniZo) Wn(z')] 



dn. 



When w exceeds unity (or when w ^ \ and ^ > 7? ^ r/o ^ 0) in (o.ll), 

 it may be verified with the help of Tables 12.2 and 12.4 that Li may be 

 deformed into L3 + L,. When n is a negati\'e integei- the quantity within 

 the brackets in (5.11) vanishes because of (4.8) and because Un{z) = 0. 

 The contribution of L3 is zero since it encloses no poles. The contribution 

 of L4 is equal to the sum of the residues at the poles gi^•en by Wn(zo) = 0. 

 Hence, when w > 1, 



E ^ —wc 



sec ,- 2^ 



2 s=i 



lA: r(n+ \)Ur.{z)Wr.{z')Vr.{z,) ' 



-/ sin x/ia]r„(so) 'a/< 



(0.12) 



where n = ih is the sth zero of ]Vn{zo). This series also converges when 

 w = 1 and ^ > 77 ^ ryo ^ (which is roughly the .shadow region). The 

 preceding inequality does not necessarily specify the complete region 

 of convergence. 



Cherry has also pointed out that the expression for a plane wave 

 given by A. Erdeha^ , namely (in our notation) 



exp [ — ix sin d -\- iy cos 9] 



sec - 



2i 



LM'^ 



+ 1) 



U„{z)\\{z') 



(5.13) 



sm TT/t 



+ Un{-z)Vn{-z')]dn, 



may be regarded as the sum of the negative of (5.8) and a similar ex 

 pre.ssion with ^ and 77 replaced by — ^ and —rj. In informal discussions 

 with the writer, Prof. Erdelyi has pointed out that tlie work loading to 

 our expression (5.5) for the field may be considcrablx' sIioiIciumI by 

 starting with some known intcgi'al for the impressed field, such as (5.13) 

 or a related re.sult. One way of doing this is to take 



exp [ — ix sill 6 + iij cos 6] + Si, 



'■ Proc. Roy. Soc Edinburgh, 61, pp. 61-70, 1941. 



