HEAVISIDES EXPANSION THEOREM 487 



The values of erf {it) for values of / from .01 to 2 are given In a table in 

 London Mathematical Society Vol. 29, 1897-98, page 519. The values 

 of erf (/g*"''*) and erf {te~'''^'*) are given by the following formula? : 



erf (/e^-H) = ^[TilC(H2/^) - iS(t^|2J^)2, (14) 



erf ite-'"'") - ^|2il- iC{t^^) + ^(/a^)], (15) 



X(\2/7r / ,9 \ 



COS [j-Ut, 



sinf^jrf/, 



C(- itAlM = - iC[t-^) 

 S{- it^Jfir) = iSit-yJlfK). 



where 



and 



and 



and 



Tables of the values of these two integrals known as the Fresnel 

 Integrals are given in various handbooks such as Jahnke and Emde. 



Examples of Application of Theorem 



A few applications of the theorem will be given. 

 Example 1: 



The operational solution for the current entering an infinitely long 

 ideal cable with a given impressed voltage of the form Ee~'^^ is 



K ^ 



P +a' 

 K being a constant and 



To evaluate p^^^/{p + a) in closed form call p^'^ = q then 



y,3/2 ^3 



p -\- a 5- + a 



Since the theorem applies in general only when the degree of the 

 numerator is less than that of the denominator we will write 



q^ aq 

 — =2 



g- + a q^ + a 



