738 BELL SYSTEM TECHNICAL JOURNAL 



In this particular problem no asymptotic solution is derivable 

 directly from the operational equation, at least by the straight- 

 forward Heaviside processes. Asymptotic solutions, however, con- 

 stitute a large and important part of Heaviside's transmission line 

 solutions. We shall therefore discuss next a problem for which 

 Hea\iside obtained both convergent and di\ergent series expansions. 



Problem B: Terminal Voltage on Cable with "Unit E.M.F." Impressed 

 on Cable Through Condenser 



We now take up a problem for which I Icavisidc obtained a di\ergent 

 solution, and which will introduce us to the theory of his divergent 

 solutions and so-called "fractional differentiation." We suppose a 

 "unit e.m.f." impressed on an infinitely long cable of distributed 

 resistance R and capacity C per unit length through a condenser of 

 capacity C,,: required the \-oltage I' at the cable terminals. The 

 operational equation of the problem is deri\-ed as follows: — ■ 



We know from the problem just discussed that the current entering 

 the cable whose terminal \oltage is V, is, in operational notation 



>] 



£Pv 



R 



But the current flowing into the condenser is 



CoPil-V) 



since the xollage across llic condenser is 1 — T. Ivtiuating llicse two 

 expressions we get 



]/= f^;___ (85) 



pCo + VpC/R 

 whicii is the operational equation of tlie problem. 

 This may be written as 



F= U- 



(85) 



where 



