Expansions of Bessel Functions. 231 



which will be referred to as the associated equation of 

 the third order. It will be recognized as the equation 

 whose solution is a quadratic function of any two solutions 

 of(l). 



Definition of the Bessel Functions. 



We shall choose, when n is not an integer, 



»i=(")* J.w («) 



y 2 = ( <r V cosec nir ( J_. H (z) — cos nvj n (z)) . . . (9) 



where, in accordance with the usual notation, 



z n / c 2 z^ \ 



J *W = 2^I> + 1)( 1 ~~2 2 7w + 1 + 2^4 2 .7 < + l.^T2" • M^ 



J -W- **i ^ 2 2 7l^ + 2 2 .4 2 .l-n.2^-]' 



• • ■ (11) 

 with the ordinary semiconvergent expansions * when ~, and 

 not >?, is large, 



(y/j.» =U^)sin(,-f + |)+V» W cos(,-f +|) (12) 



(5f)\uM-U.(,)oo.(,+ S + g-V„(.-) s in(; + f + J) (13) 



where 

 TT , , 1 4« 2 -l 2 .4n 2 -3 2 4rc 2 -l 2 .4?r-3 2 .4n 2 -5 2 .4n 2 -7 2 Mjl . 



u " (r)s=1 27W~~ + - UJSzf "- (14 > 



v ,. 4» 3 -12 4w 2 -.12.4n 2 -3 2 .4n 2 -5 3 „-. 



v *W="TTgr- 3! C8g y + (15) 



These values make 



y&i-3MI*=h (16) 



so that, for these standard solutions, C=l, -~ = ~ . 



dz R 



This definition also makes J. a (z) as (— )» J n (z) when w is 



integral. By comparison with Hankel's expansions the 

 formulae (8, 9) are obviously the most suitable for the ex- 

 pansion of (?ji y 2 ) in the forms Rl (sin p. cos p) in general, 



* Hankel, I c. 



