Roots ofBessel Function J n (a) and its denvate JV (#). 1 93 

 For the first three roots of J n (V) = 0, 



Pl = n + l-8569a> +r0344n-*..."^ 

 p 2 = 7i + 3-2447?i« + 3-1584?2-i. . > . . . (23) 

 p 3 -?z + 4'3817?ii4.5-7598 ?? -i... ) 

 From (15) and (16), it can be shown that p x is given by 

 Pt ^n+l'85576ns + l-0SZi5n-*-0'0V4:0Zn- 1 



-0'090Z'3n-? + 0-0U8n-i (24) 



All the formulae hold whether n be an integer or not. 



(c) When n is negative, X is dependent upon p, the 

 number of the root, and n, the order of the function. 

 For roots larger than n, 



and for the first root, p is the least integer making 4p— 4?i — 1 



positive. Then 



/, X 2 3\ 4 \ 

 /) = n ^l + y+ _ ...j. 



The following simple expressions give approximately the 

 first three roots of J_„(#) when n = a — -J, a being a positive 

 integer 



/3 2 = ^ + 2-596n3 + 2-02272-3 ( .. V. . . (26) 



/ o 3 =w + 3-834n*+4-41(k-3... J 



If n = cc — K and a: is less than -J, the first root is equal to 

 or less than the order of the function and the above formulae, 

 derived from the expressions when the argument is greater 

 than the order, are not applicable. The roots of J- n (n) =0 

 are given approximately by 



1 , / 36 . \* 3* T(i) 1 0-03944 



"— 6 + tex) • m ■ m — 6 + ^ziy?- < 27 > 



(d) The Function J_ n (x) possesses complex roots also, 

 which can be calculated for small values of n, from Lord 

 Rayleigh's expressions for the sum of the powers of the 

 reciprocals of the roots. If r(cos + i sin 0) are the complex 

 roots of J_i(#), 



2 cos 4(9 a.a£aao j 2 cos 80 aaa^-ia 

 z — = —0-06009 and s — =—0*00619, 



r= 1*944 and (9=61° 8'"5. 



