194 Numerical Calculation of Bessel Function J n (x). 

 When l<n<2, the first root is imaginary, e, g. 

 Pl of J._^=P"932f, of J_!=l-200z, and of J_|=l-2.61i. 



Hoots of J n '(x). 



(e) The roots of the first derivate of J n (x), — 4^- can be 



ax 



found by differentiating the first two terms of Debye's 

 asymptotic series for J n (x) and, as before, determining the 

 approximate value of e. When n is positive and large and 

 <f> a small angle. 



n(tan<£ — <p)~ — = (p— l)7r + e and ntan 3 <ft = ^ ^~ '* . 



From the first of these results, 



24cosec 2 <frtan 2 </>-5 =; 19 



€ ~ 48?itan 3 <£ 36(4p-3)<7r * ^ ' 



Hence if 



r3/ (4p-3)7r 19 Mi 



X -Ut 4 36(4p-%r(J ' ' (29) 



p p = n(l+- + — Q) ..?j i ^ = 1,2,3.... 



The first three roots of J n '( x ) f° r large positive values of 

 n can be calculated from 



/3l =7i + 0-763^3 + 0'175?i-'i..1 



p 2 = n + 2-574»** + l-988n-*... [' • • • (30) 



p 8 =n + 3-824n*-M.^387n"*.. J 



(/) Similarly, when n is negative and large, 



r-3 f (4/?-4n-3> 19 IT* 



Lnl 4 36(4p-4n-3>rJ J ' 



and as in (25), for the first root of J' (x), p is the least 



— n 



integer making 4p — 4n — 3 positive. Thus, p = 8 f or p x of 

 J'J,, and X-[«j(£ - I ^)] i - This gives 10-41 as 



