750 BELL SYSTEM TECHNICAL JOURNAL 



of having a method of checking the correctness of his processes and 

 results. 



We now take up the discussion of the asymptotic expansion solu- 

 tions of operational equations of the type 



/,=0(/,V^ (/fe integral). (123) 



In this discussion we shall, as a matter of convenience, assume that 

 iS; = o, so that the equation reduces to the form 



h=<l>iVpl (123a) 



This will involve no loss of essential generality, since the analytical 

 theor\' of the two equations is precisely the same. 



The Hea\iside Rule for this t\pe of operational equation may he 

 formulated as follows: 



// llie operational equation h= l/II{p) is reducible to the form 



h=4>(pWp) 

 and if <t> lulmits of power series expansion in the argument, thus 



h =ao+a,p'' Vp+ci'ip^'' ^^ +a-ip'^><+Wp + ■ ■ ■ 



a series soliUion, usually divergent and asymptotic, is obtained by dis- 

 carding integral powers of p, and writing 



/;=a„-f (a,/)^-|-a:i/'''^ + '+a5/'"+-+ ■ ■ • )Vp- 



The explicit series solution then results from replacing v P by 1 v""/, 

 and p" by d"/dl", whence 



(-JI)*/ 1.3. ■ ■ (2fe-l) 1.3 .. . (6fe+l) \ 



Wirt^' (20* °' (2/)=**+' ^ ")' 



a„ + 



'I'hc thiM)r\- (if tills series s()liilii>n will be based on the following 



)p<isiii<)n, (k'duclbk' h-(ini the- lilfntltN' / ^> dt = \/y/ p. 



■ .'0 Vr/ 



If the function F(p) of the integral equation 



np)= rfit)e 



Jo 



)e-t"dt 



approaches Xjy/pas p approaches zero, then f(t) ultimately behaves as 

 l/Vrt: that is, if F{p)^l/\'^as p^o, then f{t)<s,\/ \/wt as t-*oo, 

 provided thai f{t) converges to zero, atui contains no term or factor which 

 is ultimately oscillatory. 



