40 BELL SYSTEM TECHNICAL JOURNAL 



To the same order of approximation in p = w/2u m , this agrees with the 

 solution (3.18) given above. 



Appendix II 

 Properties of the Bessel Function J n (x) 



The Bessel functions have been studied and tabulated more ex- 

 haustively than any other functions largely owing to their great 

 importance and frequent occurrence in mathematical physics. Quali- 

 tatively their behavior for integral orders n and real arguments * 

 may be described as follows. 



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

 very small and positive, and is initially zero (except when w = 0). 

 In the neighborhood of x = n, 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 / 2w+l \ 



\Vx cos { x --T- ir ) 



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

 the function is as described above. 



In order to get a more accurate picture of this function the follow- 

 ing approximate formula was developed in the course of the present 

 investigation. 30 



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



where 



'.(*) = 



\tx f m 2 dm 2 1 \ >/* 



\ x 2 + 2x 4 (l-m 2 /x 2 ) 2 ) 



„ / n r I, m 2 , m . , fm\ m 2 1 ~| 



fi M (x)=x|_^l--^ + -sin-^-j-^- 4(1 _ w2/x2)3/2 j- 



2n+l 



-. T, 



nj(*) = 2£0.(*), 



f i m 2 3 m 2 1 



~\ x>^~2^(l-m 2 /x 2 ) 2 ' 

 and 



m 2 = n 2 — 1/4. 



30 It was subsequently discovered that somewhat similar formulas had previously 

 been developed by Graf and Gubler (Einleitung in die Theorie der Besselschen 

 Funktionen), and by Nicholson (Phil. Mag., 1910, p. 249). 



