984 



THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 



III. ZEROS 



When X = (2n + l)/2, n = 1, 2 , W^Jx"'"'^ reduces to a 



polynomial times a function of x- This may be inferred from the asymp- 

 totic expansion, 



1+ Z 



[m' - (x - y2)V - (x - Vz)'] •■■in'- ix-n+ y^f] 



nix"" 



which terminates if /x = 1 and x = ^i + 3^(w= 1,2...), or from the 

 fact that for these values of the suffi.xes, W reduces to the generalized 

 Laguerre polynomial 



Similarly when x = (2/i + l)/2, ?i = 0, 1, 2 . . . ., W-^^ reduces to the 

 Laguerre polynomial 



Lx-(M2){x). 



The zeros at the critical values of x are given in the following table 



Betw^een n -\- }/2 and w + M, TF^.o has n + 1 zeros (n = 0, 1, 2 . . .) 

 and TFx.i has n zeros {n = 1, 2 . . . .). 



The zeros of M^^^ coincide with those of ir^.i when x = n + 3^^, and 

 at those values of x only. The functions il/x.i and IF^.i then are propor- 

 tional to each other.* 



IV. THE RICCATI-EQUATION FOR THE IMPEDANCE FUNCTION TF^.o/^ x.l- 



The computations concerning the cylindrical helix required a study 

 of the function 



Z(«) = ^ . 



PFx.i(m) 



* At these critical values of x, the solution to the problem of the hollow cyl- 

 inder of ferrite in the waveguide breaks down, since M and W are then not in- 

 dependent. A further independent solution must then be constructed. 



