NOTES ON THE HEAVISIDE OPERATIONAL CALCULUS 153 



Now return to the operational equation (6), and expand as follows, 

 without reference to convergence, 



(10) 



We have now to distinguish three cases: 



1. Xfi > 0. (Real part of X > 0.) 



In this case it can be shown from (7) that ^ 



;z(/) ~ 5(X/) (11) 



and that the Heaviside Rule leads to a true asymptotic expansion, 

 as defined by Poincare. When X = 1, by the known expansion of 

 the right hand function in equation (9) we find that the error com- 

 mitted by stopping with any term in the divergent series is less than 

 that term. This property, however, does not characterize the series 

 for all complex values of X for which the real part is positive. 



2. X« < 0, X = - M, Mfl > 0. 



In this case, comparison of (8) with (7), gives by aid of (11), 



h{t) ~ e'"5(M/), (12) 



which again is a true asymptotic expansion. The expansion differs, 

 however, from that given by the Heaviside Rule, by the factor g''^ 

 and the alternation in sign of the odd terms of the series. 



3. Xfl = 0, X = iw. 



In this case it is easily shown that ^ 



A(/) = ^-"'-'/^>/o(f ), (13) 



where /o is the Bessel function of order zero. From the known 

 asymptotic expansion of this function, we find that 



hit) ~e-"'-'/^T^^*"'''^-5(^'c«^0]Rea.Part (14) 



with an error less than the last term included. 



* L.c. by the process described in Chap. V. 



^ L.c. formula (w) of table of integrals. Chap. IV. 



