ON CALCULATION OF MATHEMATICAL TABLES. 



319 



Bessel Function Derivative, ^ • Jv(x). 



dv 



Some twenty years ago Schafheitlin ' discovered that the Sine and Cosine integrals 

 Si (x) and Ci (a;) were closely related to Bessel functions and could be expressed in terms 

 of the derivatives with respect to the order v of the functions, v having the values +^. 

 This relation followed from the consideration of the integrals 



I sin u sin («-a;)5= /y/'y ^-^ ■ J_j(.r) + Vj(x) 1 

 and I C08M sin (u-x)^^ = Vy ["? * JiW + Wj{a:)l 



By partial integration of these two integrals 



7C 



si (2a;) = 



cos 2x 



■K 



and ci (2a;) = + -s- sin 2a: + 



-A/f[ 



cos X • Vi(a;) — sin a; • Wj(a;) 



sin X • Vj(a;) — cos x • Wj(a;) 



where si (.v) — Si (a:) — ^ and ci (x) = Ci {xj. 



Vi(x) 



8v 



Jv{x) 



t'=i 



and Wj(a;) ■■ 



Sv 



• 3,{x) 



>'=-i 



Ehminating Viix) and Wj(a;) in turn, each of these derivatives is expressed in 

 terms of the sine and cosine integrals 



I • J.(.^) 



y=i 



= Jj(x) Ci(2a;)— J_j(a;) Si (2a;) 



and 



Sv 



. J^(x) 



u^-h 



= J_i (a;) Ci (2a;) + Jj(a;) Si (2a;) 



a result independently discovered by P. R. Ansell and R. A. Fisher.^ 



Tables of Jj(a;) and J-j(a;) to six places of decimals were published in the Report 

 for 1925. 



From the relation between Bessel functions of different orders. 



J^-lW + J.'+l (x) 



2v 



Jv{x) 



by differentiating with respect to v, the recurrence formula may be obtained for the 

 calculation of derivatives of higher or lower orders. 



I •J._,(^)+| .J.+.(a:)=|! . p.(x)+h,(x) 

 ov ov a, 6v a; 



and in particular 



8 1 



8VW = x 



and — . J_3(a;) = - 

 Sv X 



2Jj(a;) 



8v 



.Ti(*) 



2J-i(a;)-|^.J-i(a;) 



8v • -^-i^^^ 

 S 



8v 



JiW 



For integral values of the parameter v. 



8v "^"^^^ 



= _G„(a;)+^!2 \a; 



n-i/2\«-P Jp(a;) 



(n-p) .p ! 



and 



8v 



■Jyix) 



= (-!)"+! 



Gn(a;)+"-2 U) . ^ , 



2 ^p=o\ / {n—p) . p !. 



Jp(x) 



