128 BELL SYSTEM TECHNICAL JOURNAL 



The expansion IjH, then takes the form 



Tl ^ z4^ " ^""^ + aie-2^-^ + f- a„e-2"^-^ + • • • ] 



+ Wan" + • • • + ak'an-k" + • • • + a„'ao"]e--"^'-^ +•••]• 

 We have then the relation 



an -= [fifo'an" + ■ • • + ak'an-k" + • • • + a„'ao"]. 



Now a„" is the value of a" when ^ = n{2D), hence the above relation 

 can be put into the form of an integral 



a{t) = f a"{T)a'{t - T)dr, 

 Jo 



where t = k/2D, and the complete relation is 



1 1 r* 



= r I r a'\T)a'{t - r)^T 1 e-'^^'dt + ao'ao". 



Suppose now that we let Z' = jco; Z" = Z(jco). We know from 

 Heaviside's rule and from Section I that l/j'co has the direct current 

 or indicial admittance solution 



i = / = h{t). 

 Hence 



g'(0 =J^h\t) =j^t=^ and Go' =0, 

 and the infinite integral equation 



J. ^ 



"W) " r^r / '""(^)"'(' ~ ^)^^i ^"""^^ + ""'^o" 



takes the form 



jwZ 



w=r[i«"(^"'']^"'-''"=r""«^"'"*'"- 



