1 205 

 and finally 



Hn ix) = — — - — e' \ e ' H„ (a) cos i — ax ] da . 



z^ 



Multiplying this equation by de 2 we have 



COS \ ~ ax I da 



ffn («) = J=: I ffn («) 6 COS \ ~ ax 



thus 



3(«2-)_x2j 



3 1 ^, . 1 10 ~ /4 w^\ 

 An = ;:;- -tJ^ (^» «) = — 73:= « COS — «^ | 



2" ' ' ' 1/5^ 



.T 



To make K{x,a) independent of n, we distinguish two cases 

 1. 71 even = 1m, then 



10 



22'V5^J ^ 5 



''/52m (^) = ^^^^^ ^ . I (f^m («) ^ COS — «A' C?^ 



3(a2-)_a,2) 



/■2m — ——. ^i {x . a) =1 e cos— ax . 



2. ?i odd =: 2m + 1, then 



(—1)'" r — ïö~~ 4 



^ V2m+i (■'^■) = 22»i+ii/F- I ''^2m+i («) ^ sin - ax da 



— 00 



(-1)- 1 '^^^=^ 4 



22'«+i 1/5^ 5 



According to the theory of integral equations we know tliat 



K,{x .a) — :E 



A 2m 



K, (X . «) =: ^ 



A2 m+1 



or 



K,{x,a) = -—e 2 :S(_i)m 



Kjr (2m)/ 



JT. (.,«) = — ." '-^ J (- 1).-. "'"'+' <"' ^""+' ^"' 

 '^ ' (/^ ^ ' (2m+l)/ 



