CIRCUIT THEORY AND OPERATIONAL CALCULUS 749 



ri'plai'e the appliwi e.ni.f. «• '^' by sin wt. The formula corrfspniulinn 

 to (120) is now 



If we now atti-inpt to expand the definite integral of (121) in the 

 same way as that of (120), we lind th.il the process breaks down Ijecause 

 each component of the infinite integral is now itself infinite. In fact 

 no asyntptotic solution of this problem exists. 



I.et lis. however, start with the operational formula: since 



Jo 



e-i" sin ut.dt^ ^ , . 



-4^ 



up 



■Vp. 



i^ + u,' 



Now expand this in accordance with the Heaviside Rule: we get. 

 o|x?rationall\', 



and expliciiK- 



/ = 



which is quite incorrect.' The incorrectness of the result will be evident 

 when we remember that the final value of tlie current is the steady-state 

 current in response to sin ut, or 



4 



^(cos a)/ + sin tjit). (122) 



This result can be derived directly from (121) by writing it as 



\~C \ r' cos uit r' sin oil t 



/ = . ^^ j cos .tl ^7^rf/+sin .tl -^di \ . (123) 



If the time is made indefinitely great the upper limits of the integrals 

 may be replaced by infinity. The infinite integrals are known : sub- 

 stitution of their known values gives (122). 



This example illustrates the care which must be used in applying 

 Heaviside's rules for obtaining divergent solutions and the importance 



' While this scries is incorrect as an asymptotic extension of the current it has 

 important significance, as we shall see, in connection with the building up of alter- 

 nating currents. 



