46 Mr Watsoji, Bessel functions 



Hence /„' (n) = - I , -p^ sin ii6 cl(b + («~-), 



TT / 1 — cos ^ ^ 



where ^, as previously, stands for 6 — sin 6. 



Now, if fs ((}>) = ^^ sin ^ . (1 - cos e)-\ then/: (0) = 2^ 3 ' ^ and 

 /^{(fi) has limited total fluctuation* in the range (0, tt). 

 Hence, applying Bromwich's theorem we have 



o f'^ sin n(6 . , ,, , , , ,., f'" sin ilr - , ,,, 



J (p'' Jo T^-- 



and so J",/ (n) = „ ^^^ + o (?i " *) + (n-'), ^ 



TTIV- 



when n is large and real ; and this is equivalent to the result 

 stated in § 1. The approximation could be carried one stage 

 further (as in § 2), but it seems hardly necessary to give the 

 analysis. 



4. As an example of the necessity for the caution which has 

 to be taken in approximating to integrals with rapidly oscillating 

 integrands, it may be remarked that some of the earlier writers 

 mentioned in § 1 assumed that when x and n are large and nearly 

 equal [in fact, when \x — n\ = o (n^)], then Airy's integral 



An («?) = -[ cos [nd - cc {6 - ^6')} cW 



is an approximation to Bessel's integral for J^ (^). This assumption 

 is correct, and it happens that the first tivo terms in the asym- 

 ptotic expansions of An (;») and Jn {oc) are the same. 



But Airy's integral for An {not) is not an approximation! to 

 Jn (no) when a is fixed and < a < 1, while n — > x . 



To establish this statement we use Carlini's formula:}: 



Jn (na) ~ -_ 



{1 + ^/(l - a2)}» . (1 _ a')i V(27rw) 



(valid when < a < 1), and after observing that we ma^- write 



An (yia) = - I — ) / cos IW (mw + lu^)] dw, 

 7T\naJ J Q '■^ ^ n ' 



* A formal proof will be given in § 5 c that /I, [cp) is monotonic and decreasing. 



t For example, the arguments given in the P/u7. Mag., August 1908, p. 274, 

 to justify the approximation seem to me to be as applicable to the second case as 

 to the first. 



X A translation of Carlini's memoir (published at Milan, 1817) was given bv 

 Jaeobi, Astr. Nach. xxx. (1850); Qes. IVerl^e, vii. pp. 189—245. See p. 240 for 

 the formula quoted. 



