54 Mr Watson, The limits of applicahility 



makes the theorems of Fejer and Hardy rather deeper than the 

 theorem of §§ 2 — 3. 



It should be pointed out that there is one integral which can 

 be regarded as coming under either head, namely*, 



/•oo 



/ X sm . 



I X " (nx + ax~n dx, 



j cos ^ ^ ' 



where n is large, a, \, and r are positive and X and r are chosen so 

 that the integral converges. [For the sine-integi-al, the conditions 

 for convergence are < X < r + 1.] As the integral stands it is 

 of the type discussed by Fejdr and Hardy, with a variable 

 stationary point where x''+^ = arjn. But if we make the sub- 

 stitution 



and then write v for n''''<»'+i', it becomes 



j,(A-i)/r[ t-^^^^'^lvit + at-'y^dt, 



Jo cos ^ ^ ^•' 



which is of the type discussed in this paper, having a fixed 

 stationary point where t = {ray/^''+^K The reader will have no 

 difficulty in deducing the approximate formula by either method. 



5. As an example of the apparent inapplicability of the 

 methods of this paper consider the integral of Bessel for Jn(x) 



when n and x are both large and a; — ?i, is (n^). 

 The integral is 



1 f'^ 

 Jn (x) =- I COS (n6 — X sin 0) dd, 



and the stationary point is given by cos = nlx; let the root 



of this equation be = /j,, and let x = n + an^ where a > ; when 

 n is large we have 



In considering | cos (n0 - x sin 0) d0, we write 



X = n0 — xsin0 — (n/j, — x sin /i,), 

 and the last integral is expressible by integrals of the type 

 »(tan^-M) cos d0 , 

 sin ^ dx 



f 



Jo 



* I am indebted to Mr Hardy for suggesting that the integral in which a- (t) = llt 

 can be reduced to an integral of Kelvin's type. 



