DIFFERENTIAL EQUATIONS 197 



9.105 For all values of v, whether integral or not: 



F " W = S75F ( cos " T/ ' ( * ) ~ J -* (x) } 



J- v (x) - cos virJ v (x) - sin virY v (x), 

 Y- v (x) = sin virj v (x) + cos p7rF,,(ff). 



9.106 For v = n, an integer: 



Y_ n (x) = (-i)F n (*). 



9.107 Cylinder Functions of the third kind, solutions of Bessel's differential 

 equation: 



1. Hi (x) = /(*) + iY v (x). 



2. Hi (x) = J v (x) - iY v (x). 



3. H- v (x) = e Hl(x). 



4. H- v (x) = e- v ^H l v (x}. 



9.110 Recurrence formulae satisfied by the functions /, Y v , H v , H?, C v 

 represents any one of these functions. 



i. CW-i(ff) - C v+ i(x) = 2-j-C v (x). 



2. 



00 



3 . 



d v 



4. -C (x) = Cr(x) - C v+ i(x). 



5 . 



6. = C, +2 (^) + C v .*(x) - 2C v (x) 



9.111 



2. 



ASYMPTOTIC EXPANSIONS FOR LARGE VALUES OF X 



9.120 



cos , - T - , * sn , - 



. F, (*) = V/ ( P, (*) sin (x - ^ J TT) + Q v 



V TTiC ^ \ 4 / 



(x) cos ^ - 



