a 



422 BELL SYSTEM TECHNICAL JOURNAL 



and, if />" is replaced by (I"ldt", 



[ 3.rl!\4// ^5.v2!V4// 



= 1 -erf-^, 



2V/ 



since the term in brackets with its accompanying multiplier is the 

 absolutely convergent expansion of the error function (erf) ; 



2 C^ 

 erf (n) = -p | exp ( — z^)dz. 



Vtt Jo 



The result may also be established either by use of an integral 

 equation •* or the Fourier integral; it is given as pair 803, Table I, in 

 tables published by G. A. Campbell.^ In the present use of the tables, 

 for unit step current, the mate of F{p)lp, where F{p) is a function of p 

 to be evaluated, is taken since the unit step function is expressed by 

 p-^ (pair 415). 



The second operational form required may be derived from the 

 first by differentiating with respect to a, since {dlda)F{p) = {dfda)f{t) 

 where F{p) and /(/) are corresponding functions of p and /. Thus, 



ayip exp ( - ayjp) = -p exp f - ^ j , 

 since 



^^erf IHOI =^^'(0 exp | - IHDJ } ' 



The unit step voltage may now be expressed, by substitution of 

 these results, by the following formula: 



2r^/-exp( --y- 



dSds. (1) 



In equation (1), as in the steady-state formula from which it is 

 derived, the wires are unrestricted in path or length on the surface of 



* J. R. Carson: " Electric Circuit Theory and The Operational Calculus," McGraw 

 Hill Co., 1926, p. 19, eq. 29. 



* "The Practical Application of the Fourier Integral," Bell System Technical 

 Journal, October, 1928. 



