'=^iwl 



CIRCUIT THEORY AND OPERATIONAL CALCULUS 743 



(operatiimal ec|uation), viz. in rising powt-rs of (>." Thus, rt-tiiniiiiv; 

 111 (!•'.)). it I'.in li«- wriltiMi as, 



\ L y/l+P/2\ 

 Now i-xpaiul the ilciiominalor by thf binomial liu-urrni: \vc ^-i 



/ = J^: .; i_A+l£VAy_LMf MV . . ' 1 ^ (105) 



' \l ' 4X^2!V4Xy 3!Ux>'^ '\2X 



He now identifies v'p, 2X with 1 y "-rrX/ and rcpkucs /j" in the series 

 by d'dt', thus getting finall>' 



V2k\I ' ^8X/^2!(8X/)'^3!(8X/)'^ ' ' ' » 



This series solution is divergent : Heaviside recognizes it, li()we\er, 

 as the asymptotic expansion of the function ^"^'/..(X/), and thus 

 arrives at the solution 



/=^)^\-X'/„(X/) (101) 



which we have obtained from our tables of integrals. 



Now the divergent expansion (100) is the well known asymptotic 

 expansion of the function e"^'/„(X/), which is usually derived by diffi- 

 cult and intricate processes. The directness and simplicity with 

 which Heaviside derives it is extraordinary. 



We note in this example that no integral powers of p appear in the 

 divergent expansion: consequently no terms are discarded. Other- 

 wise Heaviside's process is as startling and remarkable as in the 

 example discussed in the preceding section. 



We shall later encounter many problems in which asymptotic 

 st)lutions are derivable as in the preceding example. We have suffi- 

 cient data, however, in these two typical examples to take up a 

 systematic discussion of the theory of Heaviside's divergent solution 

 of the operational equation. 



CHAPTER V 

 The Theory of the Asymptotic Solittiox of Oper.\tional 



Eyi ATIONS 



A study of Heaviside's methods, as exemplified in the preceding 

 examples and in many problems dealt with in his Klectromagnetic 



