147 



— 00 



If now the conditions 



are satistied, /ir(A') must be reducible to 



e-^ [ajl, (..) + ajl, (,r) + a,H, (.r) + . . .] 



It is easy to show, that this is the case ; for if the conditions are 

 satisfied we have 



thus 



00 



CO ^ 



CO 



or, introducing (9) 



//2A;(A-)e-^'=:^-— ^ I e /2A: C06- ;.« C?/ 



K jr J 



CO 



00 >^ 



{—l\k f —- 



H2k-\-i (.v) é-^' = —-=- I e A2A:+i 5ï« ;.r (f A 

 K{w) = e-' [a, /ƒ„ (.f) 4- a, i/„ (w) + a, ƒƒ, (^0 + ...] . 



17. From the relation (32) another important result may be 

 deduced. For multiplying by e—!/^ dy and integrating between — oo 

 and 00, this relation gives 



00 



or, putting 



X + y=: «1/2 



00 



J (1/2")"+^ 



00 



Therefore, assuming 



