Equal Order and Argument. 



233 



(II.) Dr. Airey's formula (2) seems to me to be mis- 

 leading. In it he equates the Bessel function J«(^) to an 

 asymptotic expansion which is not that of the Bessel 

 function but is the asymptotic expansion of Airy's integral ; 

 it is well known that this integral is only an approximation 

 to J„(r), and, in fact, the asymptotic expansions of J n {z) 

 and of Airy's integral agree in their first two terms. This 

 fact appears to be implied by Dr. Airey's statement : " The 

 third and following terms are correctly given in (14)" ; 

 I think that a rather more definite statement would have 

 been desirable, but this is a comparatively trivial matter. 



(III.) In finding the asymptotic expansion of J n (n), 

 Dr. Airey uses Bessel's integral, and proceeds thus*: 



1.00= -( 



7T j 



cos { ?i (sin w — w)} di 



"»00 



1 C f /u; 3 w 5 



=A cos t"U-i2o 



+ 



w' 



5040 



and gives a reference to Lord Rayleigh's wor 

 Dec. 1910). Now the last expression is really 



. ) v die, 

 (Phil. Mag 



7i 1 



and this last 

 divergent ; at 

 to be the case 

 we have 



7T J 



■2kr 



COS 



cos \n(w—sm w)} div, 



*y'0 



integral is not even oscillatory but is quite 



any rate when n is an integer, as is supposed 



r or, if we take k to be a (large) integer^ 



k i 7r 

 {n(w — s\niv)}di/:=— \ cos \n{w — sin w) \ dw, 



7T J _ 



because cos {n(iv — sinw)} is a periodic function of w with 

 period 2tt ; since, in addition to being periodic, the function 

 under consideration is an even function, we have 



I cos \n(io — sin to)} dw = 2 \ cos {n(w— sin w)\ dw, 





div — 



cos {n(w — sin w)} dw 



and so 

 — \ cos {n(iv — sin w) 



= 2AJn(«)i 



and this obviously tends to infinity with k. 



* It is implied that the infinite integral is an approximation to J n (n). 

 The fact that the exact expansion (11) is derived from this result which 

 purports to be merely approximate indicated to me the desirability of 

 making this investigation. 



■ 



* 



