Chessin — On Some Relations Between Bessel Functions. 105 

 (19), x^ \^Jn{ax)K„(ax)—J,_^{ax)K,^,(ax) | 



= 2 I xJn [ax)lLn {ax)dx + Const., 



(19)3 x2 I Kl{ax) — K,,_^{ax)K^^^{ax) l 



r 2 



= 21 xKn{Q'X)dx + Const., 



when /3=a. Both sets of formulas (18) and (19) refer 

 to the case of an integral n. When n is not an inteo^ral 

 number these formulas should be replaced by the following 

 ones: 



(20), x^aJ^{^x)J,,^,{ax)—^J^{ax)J^^,{^x) | 



= (a2 — /32) I xJn(ax)Jn(^x) dx -\- Const., 



(20), X I aJ_^(^x)Jr^,(ax) + ^J„ (ax)J^+, (^x) | 



= (^2_a2) I xJ'„(aa;)J_„(/8x)c^x+ Const., 



(20)3 X I aJ_„(^x)J_„+,{ax) — ^J_„(aa;)J_„+XA^) } 



= (a2_ ^2) I xJ_^(ax)J^(^x)dx + Const., 



provided a ^ ^. If /3 = a, the last relations assume the form 



(21), x"^ I Jl(ax) — J„_,(a:7;)J'„+,(a.'«) | 



'' (ax)c?x + Const. 



= 2 aJe/n 



