GUIDED WAVE PROPAGATION Til HOUGH GYKOMAGNETIC MEDIA. II 985 



We Avill show that Z^(u) satisfies a non-linear first order differential 

 equation of Riccati-type. From the difference relations, equation (36), 

 we have 



say 



and from Whittaker's equation 



Therefore 



or 





;;^ ^ + -, + M i - 2x ) + (x^ - K) = 0, 

 dug g- g \u 



l = ^ + S-2^)^+(^'->4)^' 



Finally, let 



Then 



dZ_ 

 du 



z= ix- K) giu) = 5^) 



(x - 3-^) + (^ - 2x)^ + (x + M) Zl 



Since this equation is satisfied by Mx.oiu)/Mx,i(''^)) as well as by 

 Wx,o{u)/Wx,i(u), a selection has to be made from all the possible solu- 

 tions of this equation. We require the one which for large u approaches 

 unity. But for large u the equation is 



^ = {I - z) [x - y2 - (x + y2) z], 



whose integral is 



^ ^ (x - M)^g" - 1 

 (x + 3^)^e" - 1 



For large u, therefore, the solution is either unity, when A = 0, or else 

 (x — K)/(x + 3^), A z|z 0. The solution with A i^= corresponds to the 

 M functions; that A\ith A = to the IF-functions. 



