1201 



where according to (4) 



H,; [y) = 2n iJ„_i {y) 



H,;' (y) = 2-^ n (n - 1) H„_, (y 



thus 



H^"^ (y) = 2" . nf H^ (y) 



Hni!ri-ii)=HAy)-\-^2ni^H„_, (y)-{-2-'n{n-l)^^H.,__,(y, ^ . ^^2" .nf^^ H^{y). 

 Iiitroduciiio- tliis value, we get immediately 



An = -^- e~>/\h, = 'i 



GO 



and 



, .v.-^' - 1 + ;^, H. (,.) + 1 H, ,.,,.) + g H, ,„.,.+ . . . 



Fj'om lliis equntioii several olliei-s may be deduced, lor inslaiice 



' -'""'' = ' - 17 ''■ <••■» + £ ^' (•^» - £ '^. ("■) + ■ • ■ , 



e2/S2- — e— 23.1- J2 o4 



V^"-*)- 



III. As a third example we will expand a discontinuous fundioii. 

 Supposing /{.t) = -l from .*■ = to ,v =: \ and /{.r) = () for 

 1 <^ ,/■ <^ 0, we have 



where 



1 1 C^ 

 An — ; --: e—' H,, [a) 



da. 







This coefficient may be determined in the followiug way. 



Let . ° ' 



/„ =je-*'' H„ {u) da =j {latin--, - 2 (;.- I) //, ,) e—'da 



tlien 



78* 



