206 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



9.225 If v is half an odd integer: 

 sin 



/- t 



2\/ X 



r (V) - d Ci(x\ - sin (2 V* + e) cos (2V5 + e) 

 ~ C *. ( * ~^i~ -^ -- > 



r ( \ d r ( \ 3-4* . f r \ \ 3 cos (2\/a + e) 



|W = ~ ^ = ~~~ Sm ^^ + C) " -2 -' 



= -cos (2v^ + e), 



e is arbitrary so as to give a second arbitrary constant. 



9.226 For x negative, the solution of the equation: 



d?y f x dy 



x +(r+ 1 )JL-y.o, 



when v is half an odd integer, is obtained from the values in 9.225 by changi 

 sin and cos to sinh and cosh respectively. 



9.227 



(m + n + i)jC m+ i(x)Cr+i(x) dx = - xC m+l (x)C n +i(x} - C m (x)C n (x), 



(m + n+i) Cx m + n C m (x)C n (x) dx = x m + n +i ( xC m+ i(x)C n+l (x) + C m (*)C B (*) ] 

 v I J 



9.228 



* cos 2 <> rf 



2. * cos 



3. / C (x sin 2 <^>) sin 



4- PCi (* sin 2 0) sin 3 </> J</> = Q 0) . 



l".,, 7 . ,v . , , I COS 2-\/X 



5. ! Ci(* sin 2 6) sm 



*/o 



