NOTES ON THE HEAVISIDE OPERATIONAL CALCULUS 155 

 Now set ^ = 0; from (16) we have 



r( 



"-^+14'^'=-- '''' 



a formula which again is valid by reason of the restrictions imposed 

 on l/pH(p). 

 Proceeding in this manner we get the formula 



Jo 



(h - Sn)-t-dt = (- l)"w!a2n+i, (23) 



'0 



where 





+ (- 1)"1.3 ••• (2« - 1) '^'" 



(20" 

 = first {n -\- \) terms of the divergent Heaviside series. (24) 



Also since 



S.» = 5. + (- !).+■ '■'•;3//.: + '^ ^ (25) 



we have from (23) by changing w to (« + 1), 



f 



Jo 



h-s.- (- i)"+^ ^-^"(J^!: + ^^ ^)/"+w/ 



= (- iy+'{7i + l)!a2„+3. (26) 



Equations (23) and (26) establish the fact that (h — 5„) converges, 

 for indefinitely great values of t, at least as rapidly as l/t"+^-y[t, since 

 otherwise the integrand of (26) would diverge; stated in mathematical 

 notation 



h - Sn= 0(1/ 1"+^'^). (27) 



Consequently the series S when divergent is a true asymptotic ex- 

 pansion, as defined by Poincare, of the function h. 



The foregoing says nothing, it will be noted, regarding the error 

 committed when 5„ is employed to compute the function h. Nothing, 

 in general, can be said about this question, which requires an inde- 

 pendent investigation in every specific problem. In some cases the 

 error will be less than the magnitude of the last term of Sn, but this 

 is the exception rather than the rule. In other exceptional cases the 

 series may even be absolutely convergent. 



