GUIDED WAVE PROPAGATION' TIIHOTTGII G VHoAIAGNETIC MEDI.V. II US.S 



?/i(x) and W2(x)- There are now two possible modes, with the same 

 restrictions on p,i . An additional mode arises each time x is allowed to 

 patss a number of the form (2n + l)/2. It is to be noted that those 

 modes are not confined to the resonance range. For /3 positive, they can 

 exist in the range <x > o- > ao and in the range — o-q < o- < 0. 



Appendix I. Some Properties of Wiiittakek T^tnctions Used in 



This Paper 



I. relation to bessel functions 



if/-!^i^ = 2'«r(M + DUr), 



Lit rp— — / — , 



x-»o v2a; Vx 



y ^ X^A^ -^1^) V TT -M+lrr (1)/ \ j 



V2jx V -2jx J 



Lt 

 x-o L 



II. difference relations 



The foUoAving results can be obtained either by reference to Erd^lyi, , 1 

 pp. 258, 254, by differentiation and subsequent integration by parts 

 of integrals such as 



u+l/2 -x/2 .CO 



r(M + H - x) •'o 



or by observing that combinations of the form 



satisfy Whittaker's equation with m "= 1- In the last mentioned method, 

 the required constant multiphnng the first order Whittaker function 

 can be obtained by reference to the limiting behavior for small x. If 

 ■^x^ = ^^xA^)/y/x and *SxM = My^,^{x)/\/x the results are 



i^xo' + X^xo = (X - yz)Rxl , 

 >Sxo' + x5,o -V2{K- X) >Sxi . 



