1195 



Ln {x) 



(-1) 





(-1) ^ 



e ** w" sin ,vu du 



.,2^.+l 



^ ^ (2/;+l)/J 



or 



(-1)2 00 /n + 2m , (2a')2^+' 



ü ^ H 2 ; (2^+1)/ 



1/ 



jr 



(—1)'" 



2.t— (m + l)'^-|-(m-hl)(m ^2)l^ 



Proceeding in the same manner if n is odd, we get 



l-(;« + l)'^-i-(m+l)(m + 2)^. 



(_l)m+l 



Z2,„+i(.t;)==^— 22'«+im/e^^ 

 1/ rr 



(U) 



(15) 



Both these series are converging for all finite values of the variable, 

 and show that 









V\ 



(n even) 



(?/ odd) 



(16) 



5. Te investigate the value of Ln [x) for large values of x, take 

 the differential equations 



dx' 

 d'Ln 



2x —^ r\- 2n Hn = 

 dx 



dLn 

 — 2x — - + 2h L„ = 0. 

 dx' dx 



Multiply the former by />„ the latter by /ƒ,, and subtract, so: 



d'Ln d^Hn 



dx^ dx^ 



dLn dlln 



2x (/ƒ,"_ Ln ---1 = 

 dx dx 



or integrating 



dx dx 



C, being the arbitrary constant. 



Introducing tiie relations (4) and (13) (his may be written 

 2niHnLn^x- L,,Hn-x\ = C^\ 



