LAMINATED TRANSMISSION LINES, I 945 



n-o n! 



KoUpi) = KoUpi + kO = 2-/ — 7- -^o" (kpi), 



ri=o n! 



/i(kP2) = hixp-i) = X^ — - /o" ^ (kpi), 

 n=o n! 



(A2) 



/m(k-p,) = -Koixp,) = -E ^ Al"+^\.pO. 



„=o n! 



It follows that 



where 



«> (A" 

 Ko{Kpi)IliKp-2) + /Vi(kP2)/o(/vPi) = — 2J r ^n+l(«Pl)) 



°° f -/")" 

 7\o(k-Pi)/o('vP2) — Ku{KP'>)Iit{Kpl) = —22 r~ '^"('^Pl)) 



n=o n! 



/Vi(/CPi)/i(kP2) — A'i(kPo)/i(k'Pi) = 23 r ^n+l('^Pl)) 



00 /An 



/Vi(kpi)/(i(kp2) + Iu{kp2)Ii(kpi) = 2^ — r- ^nC^'Pl), 



A„(.r) = Io(x)K',"\r) - Ko(x)n''\x), 

 B,,(x) = /o(.r)/vr'(.r) - A'o(^-)/o"^(:r). 



(A3) 



(A4) 



The quantities .4„(.r) and B„(.r) turn out to be finite polynomials in 

 1, .r, the general expressions for the coefficients having been derived in 

 a rather inaccessible monograph by Pleijel. When x is large, however, 

 the leading terms are quite simple. From Pleijel's analysis, or directly 

 by substituting the asymptotic series for Io(x) and Ko{x) into (A4), we 

 find 



AUx) = 1/x + 0(l/x'), 

 A2m+i(x) = -mix + 0{\lx), 



2 4 (A5) 



Bim{x) = vi/x + 0{\/x ), 



where m is a positive integer or zero. 



If we substitute these approximations into the first of equations 

 (A3), we obtain 



2 H. Pleijel, Berdkning af Motstand och Sjdlfinduktion, K. L. Beckmans Bok- 

 tryckeri, Stockholm, 1906. 



