94 BF.LL SYSTEM TECHNICAL JOURNAL 



The function 7„(.v), however, is oscillatory and ultimately behaves as 



I 2 / 2«+l \ 



For all orders of w 



./o 



The properties of Jn(x) may be described qualitatively as follows: — ■ 



When the argument is less than the order (0^x<n) the function 

 is very small and positive, and is initially zero (except w'hen « = ()). 

 In the neighborhood of x = 7i, the function begins to build up and 

 reaches a maximum a little beyond the point x = n. Thereafter the 

 function oscillates with increasing frequency and diminishing ampli- 

 tude, and ultimately behaves as 



I 2 / 2« + l \ 



When n = 0, the initial value is unity, but the subsequent behavior of 

 the function is as described above. 



A more precise description of the function is gotten from the follow- 

 ing approximate formulas. 



J„ (x) = B„ {x) cos 9,,, {x) , for x > n 

 where 



\7r.V 



V le~2x' {i-myx^yj 



^ , , r li "^'" , m . .f m\ m- 1 n 2w+l 



n;,{x) = -^iUx), 



_ I m^ 3 m^ 1 



and 



x- 2 x' {l-m^x'^y-' 

 m- = n-— 1/4. 



This approximate formula is \alid only where x>n, its accuracy 

 increasing with .v and with ;/. For all orders of n it is quite accurate 

 bevond the first zero of the function. 



