152 BELL SYSTEM TECHNICAL JOURNAL 



If the operational equation h — \/II{p) admits of formal series 

 expansion in the form 



h = aa -\- aiVp + a^p + a^p^p + a^p- + . . . , (4) 



a solution, usually divergent and asymptotic, results from discarding the 



/— ^" 1 

 terms in integral powers of p, and replacing />"\/> by -7— — -^ , whence 



;.'^ao + |ax + a3|^ + a.|^,+ ---}-i=- (5) 



As stated in a forthcoming paper, this divergent series is a true 

 asymptotic expansion, as defined by Poincare, if and only if, the 

 singularities in 1/H(p) all lie to the left of the imaginary axis in the 

 complex plane. Otherwise the series may require the addition of 

 an extra term or factor, or even be quite meaningless. 



An excellent illustration of the preceding principle is furnished by 

 the operational equation, 



V/> + X 



For convenience and without loss of essential generality we take 

 |X| = 1 and X = e*^; that is, the parameter X may lie anywhere on a 

 circle of unit radius in the complex plane. 



Now the solution of (6) is easily derived by well known processes 

 of the operational calculus: it is 



h{t)=- \ ' dr (7) 



^ Jo Vr-V/ - r 



^ Jo ^T^r^ 



= dT. (8) 



T 



The solution is also known to be '^ 



h{t) = e-O^^l'U, (^], (9) 



where /o(X) is the Bessel function Jo(ix). 



a true asymptotic expansion. On the other hand Heaviside in his frequent appli- 

 cations of the Rule gives no hint or indication of the restrictions imposed on its 

 applicability. Fortunately in most applications of the operational calculus to physi- 

 cal problems, the Rule leads to correct results. 



^ See formula (p) of the table of integrals in Chap. IV, "Electric Circuit Theory 

 and Operational Calculus." 



