430 

 where 



BELL SYSTEM TECHNICAL JOURNAL 



Vx2 + «2 r 



exp 





exp (iw/ - tVa^ + ^2) erfc ( A/y (-"^^ + «^) - V/w/ 



ttX 



+ exp (7a;/ + tV.V'' + w2) erfc (^y (.v^ + ^/^j + Vn^ 



too ex 



X 



picot I exp ( - iwt - ^^— ^ j erf ( ?/ a/— ) ^/Z- 



The integral appearing in <J>(?0 apparently cannot be expressed in 

 closed form in terms of known functions; for numerical results series 

 or asymptotic expressions may be derived but it appears more desirable 

 to employ numerical or mechanical integration using the unit step 

 voltage since tables or charts of the error function of complex variable 

 which also appears in <J>(m) are not available. 



A useful check on the above formula is obtained by taking the limit 

 for / = 00 , which gives the steady-state mutual impedance between 

 straight parallel wires; the result is as follows: 



Zi2 = En{t) exp (- iut) 



= ttV [^(22 + fl) - ^(S2 - a) - ^(si + a) + ^(2i - a)], (5) 



where 



^(zO = 



-^x^ + n^ 



x^lx^ + u'^ 



exp (— y^x"^ + «^) 



--f 



■'■Jo 



exp I — w 



Yx^ 



eri — j= aw 



Vx^ + u'^ 



+ 



Vx^-f- 7('' 



1 — exp ( — 7 V.v' + ir) 



y" r r I 



7 I exp ( — 7 V-x^ + ii^)dw, 



where as before 7^ = 47rXfco. 



The third term in ^{ u) approaches the limit given because 

 erfc (— ^f^^) = 2, erfc (Vi 00) =0; the integral term as given in 

 the first form of ^{u) has been transformed by the substitution 



