242 Dr. J. W. Nicholson on the Asymptotic 



where X is large, and /, F are uniform in the range, is 

 represented to an order X" 1 by 



I=( f(0)e-Wdt, .... (60) 



Jo 



orhj l^Ctf'tyr+Wdt,. ... (61) 



Jo 



according as / (0) is not or is zero. Moreover, ¥(t) may be 

 expanded, and the first non-vanishing power of t alone 

 retained, if F(i) does not contain X. Also F(0)=0, and 

 F' must not vanish in the ranges except perhaps at £=0. 



In the present case /x= - — ^— r? an(1 n is ^ ess tnan z -> so 



that F ; as v f cannot vanish. The other conditions are obviously 

 satisfied. 



Now when * = 0, v = 0, and \-,t ± ) ~i ls zer ° when r is 



' ' \y dtj v f 



odd, and when r is even, it is given by (f„X r+1 Ur+i, where 



u r+1 represents the term in R next after the final one retained 



(in the second expression for R) with n — x substituted 



for ~t ~» ^. and the summation and integration prefixed. 



Thus the remainder after r terms only differs by an order 

 X -1 from 



2 l dyjru r+ i \ v e~^ v dt, or 2 I Xu r+1 —, 

 Jo Jo Jo * 



when r is odd. When r is even its order is less. 



If the rth term of Rbe therefore denoted by ir\] r (z 2 — n 2 )~*, 

 where Ur is a certain operation, the error involved in stopping 

 at the rth term is of magnitude 



- u, + i J^ df [ zcoshyjr _ n + - oS -j~), 



1^,-i^, ..... (62) 



and is therefore of the same magnitude as the term next 

 after the last retained. The expansion of R is thus asymptotic 

 in the proper sense. That of p will also be asymptotic, but 

 less convergent. 



