488 BELL SYSTEM TECHNICAL JOURNAL 



and 



q _ Y{q) _ F(0) ^ F(gJ 

 g^ + a Z(g) Z(0) ^ f g„Z'(5„) '^^^"' ^' 



F(0) _ Q F(g„) _ gn _ 1 



Z(0) ■ qnZ'(qn) 2qr? 2qJ 



Z{q) = q- + a and the roots of Z{q) — are g™ = ± m'^^, 



^(?n, /) = e^"'!! + erf g„/i/2j (see equivalent No. 2). 

 So 



?1r^ = ^ '""^^ + "'■^ (mi/¥/^)] - 2^^ .-[1 + erf (- ia^'H^i^)!, 



= ^e-«'erf (mi/¥/2). 

 Hence 



-^rV- = 5 - -1-X— = F7T7T ^ ^ ^rf (ta^'H"^), 



q^ -[- a ■" q^ -\- a Ti^i) * 



since 



^ \l/2 1-112 



= ^1/2 = _ = 



q = p 



dtj r(3^) 



Example 2: 



The operational expression for the current entering at time Mn a 

 cable of distributed resistance R and capacity C with an electromotive 

 force sin wt impressed is given by 



where 



Put q = p^i\ Then 



pZI2 _ g3 



p2 ^ ^2 g4 _J_ ^2 



Here 



F(0) _ F(g„) _qn' _ 1 



Z(0) ' 2„Z'(g„) 4g„-' 4g„ 



If 



g4 ^ ^2 = 0, qn = coi/2ei(W4)^ u^i'^e~^^^l'^\ - a)i/2e<W4)^ _ ^oi /2e-*('^/4) . 

 So 



