Bi 



DIFFHACTIOX OF RADIO WAVP:S BY A I'AHAHOLIC CYLINDKR 503 



,— 2/3 r ^oocxp (t2?r/3) 

 !t/3) 



3 



— 2/3 p |.oocxp(t2?r 

 ^TT [_ J« fxp (— i2« 



•'« exp (— ''- 

 TT Jo 



)r/3)J 



exp [-I'xs - s73] ds 



(13.16) 



exp 



,3 



-~ + xt\ + sin f ^^ + a;/ 



dl 



is also tabulated in Reference 11 where it is shown that 



Ai{xi"') = f%ii{x) - Wi{x)]/2, 

 Ai{xi^*") = r'"[Ai(x) + iBi(x)]/2. 



(13.17) 



With the help of these relations we may evaluate the expressions (13.14) 

 for Un{i 'p), etc., on any one of the six rays 



arg (m - ip-/2) = ±57r/6, ±/7r/2, ±t/6. 



When 6 is a general complex number the expressions (13.14) may be 

 evaluated with the help of the modified Hankel functions hi(a), h2(a) 

 tabulated in Reference 27 for complex values of a. The relation needed 

 is 



Ai(a) =l-h,(,-a) -^±Ji,{-a), 



k = (12)'"{-"\ 

 When I arg a | < tt we have the asymptotic expansion 



Ai(«) ~2-V"''V' (exp [- (2/3)a'^'1)(l - 5/48a:''' + • • 

 and when ] arg (— a) \ < 27r/3 we have 



(13.18) 



) (13.19) 



-1/2 



\-l/4 



Ai(a) - TT-^'i- a)-"* shi [(2/3)(- aY" + 7r/4] . (13.20) 



Both of these expansions follow from the discussion of the asymptotic 

 behavior of hi{a) and h2{a) given by W. H. Furry and H. A. Arnold.^^ 



Asymptotic expressions for Unii'^'^p), ■ • • valid when n is near —ip/2 

 may be obtained by applying the relations Un{z*) = [Un*(z)]* ■ ■ ■ given 

 by (9.11) to the expressions (13.14) for Un(t''p), • • • : 



(13.21) 



"Tables of the Modified II:iids;ci Funcfioiis (jf Order One Third and of Their 

 Derivatives, Harvard Univ. Press, 1945. 



