Mr Watson, The limits of applicability etc. 49 



The limits of applicability of the Principle of Stationary 

 Phase. By G. N. Watson, M.A., Trinity College. 



[Received 22 November 1916.] 



1. The method of approximating to the value of the integral 



». = ^r- cos [m {oD — tf(m)}] dm, 



Ztt .' 



where x and t are large, by considering the contribution to the 

 integral of the range of values of m in the immediate vicinity 

 of the stationary values of m {.r — tf(m)], is due to Kelvin*, though 

 the germ of the idea may be traced in a paper published nearly 

 forty years earlier by Stokes f. 



Kelvin's result is that, if m{x — tf(m)] has a minimum when 

 m — [x> 0, then, as ^ — > oo , 



u ~ i^-nt) - * {- ,xf" (/.) - 2/-' (fx)] - * cos [t,x\f' (/x) + iTr} ; 



and this result has imjDortant applications in connexion with 

 various problems of mathematical physics J. 



Kelvin, in his analysis of this interesting asymptotic formula, 

 takes for granted, on physical groimds, the validity of a certain 

 passage to the limit. This process requires justification from the 

 purely mathematical point of view ; and the necessary justification 

 is afforded by a convergence theorem due to Bromwich§. This 

 theorem plays the same part in dealing with integrals as an 

 analogous theorem, due to Tannery |j, plays in connexion with 

 series. 



The special form of Bromwich's theorem, which is required in 

 the rigorous investigation of Kelvin's theorem, may be enunciated 

 as follows : 



If f{x) be a function of x with limited total fluctuation in the 

 range x ^ 0, and if 7 be a function of n such that ny —^ 00 as 

 n —^ 00 , then, if — 1 <m<l, 



* Phil. Mag., March 1887, pp. 252—255 {Math, and Physical Papers, iv. 

 pp. 303—306). 



t Camh. Phil. Trans, ix. (1851), p. 175 (Math, and Physical Papers, 11. p. 341). 



t See Macdonald, Phil. Trans. 210 a. (1910), pp. 134—145. 



§ Bromwich, Theory of Infinite Series, p. 444. In the special case 7H = 0, which 

 is explicitly considered by Bromwich, the result is important in the investigation 

 of Fourier series by the method of Dirichlet. The theorem given by Bromwich on 

 p. 443 is equally applicable to the more general case. 



II Fonctions d'une variable, p. 183. 



VOL. XIX. PT. I, 4 



