368 BELL SYSTEM TECHNICAL JOURNAL 



the integrals diverge if the attenuation is not zero the use of an infinite line 

 formula presupposes reasonably efficient transmission. 



Since 11 a e~'^' we have 



i(iik~ — Ila; = - III = — Zi2 = — Zi2 . 



dX k ik 1(j3 



Since — — = r^ or iki = iojQ or I = cQ this is 

 dx ot 



tcor'l^n, = Qc' [2 log p"/p' + 4(0 - iP)]. (5) 



We have next to consider 



Faz ' ik^ dzLJ dxh (I + m)(l -{- r^m) ^ 



The infinite integral is 



Jo (/ + ni){l + T^m)[_ -00 J-oo J 



Since Jo{v \/x^ + y^) and cos kx are even functions of x and sin kx is an 

 odd function of x 



/ Mv yjx^ + y^)e~'^'' dx = 2 j J^iyy/x^ -\- y^) cos kx-dx 



= if V <k 

 ^^cosW^l if ;fe<. 



and so our integral is 



/•OO ~vol J 



^^^ h {l + m){l + T^w) ' ~7~ '^'^ 



or, since V- — v^ — W-^ 



r* g-"^^ co?> yl'dl 



^' io (/ + \//=^ + t2(/fel - k')){l + rV/=^ + i^kl - ife2)) 



or, if we put / = v y/a 



