Expansions of Bessel Functions, 239 



Thus when n is an integer, 



ff ^- J9 W (i#) = 21 <f> sin (to sin <£ — 2ra</>) d<£ 



-2?r j dcj)e- 2n< l , - W3inh< P, 



so that on reduction 



Again 



2J n ^P =| 1 cos 2/i0 H (2* sin 0) d<£- 2 1 </> sin 2»</S J (2s sin <£) ^ 



where H (w) is Struve's function * defined by 



2 f 7r 

 H (w) = - I sin (to sin (9) d<9. . . . (48) 



t t f *" t /o • \ sin 2; '« 1 ' ; 



■J.J-.=| Jo (*■«) snsF** 



so that when ?* is integral, 



/ T BJ-« L/ NnT BJn\ f^T/o • , <* sin2vwr 

 ^(Jn-s K — ) J«X =1 J (2: sin r . — -T— dx 



f^ t /n - x (-•*' cos 2h# sin rnr — TT cos tjtt sin 2w.r) , 

 = 1 J (2~sm^) r-5 -dx. 



Jo Sin J 7,7T 



The integrand takes the form : when n is integral. 



Therefore evaluating in the usual way, which is obviously 

 legitimate, we obtain 



_)»j n <Wz» + J n ^L = * f" (w«-4jfl sin 2n* J (2; sin *) <**. 



By subtraction with (47), and with the help of the results, 

 true for integral values of t?. 



(50) 



f sin2n«J (2«sin*)rf*=0 (49) 



x 2 sin 2 Wcf J (2^ sin x) dx = ir\ x sin 2nx Jq{2z sin a;) d#, . 

 ~o Jo 



it appears that 



k(*)Y„(*) =i ("cos 2„., H (2; sin .)&- * f " A» f " ^ a- 2 "'- 2281 "" sinh * 



o « o Jo . . 



(bl) 



* Cf. Struve, Wied. Ann. Bd. xvi. 1882, p. 1008; Lord Raylei"h, 

 * Theory of Sound,' § 302. 



