Functions of Equal Order and Argument. 523 



Reversing this series, we get 



Hence 



Z 1 CO 



But I ^- 1 cos^^=r(p)cos^ (10) 



Therefore 



1_ 



\/3 



j -w-^K^G)-i©* r (i)-^o© ,r (» 



+ 18 



,4»©'K1) ■]• <"» 



The coefficient of the last term in the bracket, T^mja?)' 1S 

 approximate. 



Even for very small values of n, this formula gives J n (n) 

 with considerable accuracy. Ji(l) is given correctly to 

 three places of decimals, viz. 0*43995 instead of 0*44005. 

 A closer approximation is obtained as the value of n increases, 

 J 6 (6) to six, J 48 (48) to nine, and J7so(750) to fourteen places 

 of decimals. 



n. 



6 



0-2458 



37 





48 



0-1230 



7185 7 





750 



0-0492 



3244 5583 



97 



Although n must be an integer in (8), the formulae 

 (11), (13), and (14) appear to be valid even when n does 

 not fulfil this condition. 



When z and n are nearly equal, say z = n + /c, 



1 f °° 

 ■J n (z) — — I cos [^(sin w — w) + kio] dw 



if 00 . if 00 .. 

 = — l cos r (sin w — w) cos kw dw 1 sin £:(sin w — w) sin /cwdw. (12) 



"Jo "Jo 



