148 



we obtain 



yH(a;) = - — — _ — I e - ^ 3 J(fn{ct)da 



V jt J 



00 



thus, in the same way as in Art. 9 



-I 3 / 4 - \ 



- , 1 ( a2 a.r ^/2 4- aM 



A„ =: (l/2)«+i, K{x,a) = -= e 2 I 3 ^ A 



Here the value of the function K{x . «) is unite for .I'and « ± oo. 

 In the same manner as in Art. 9, therefore 



K{x,u) =z > 



/« 



or 



• , . - 1 ^^ 



2 2 ,i/ 



which may be verified by (9). 



18. Now, according to the theory of the integral equations the 

 determinant D{X) of the kernel K{.v,a) must vanish for the values 

 ;. — (|/2)"+i(^i = 0, 1,2 . . .). 



To examine tliis, we write D{}^ in the form which is given by 

 Plemelj ^) 



where 



— («i-l-«,^-4-«3^'+ • • ) 



CO CO . 00 



ttj =: I K(x,x)dx, a^= I K^{j!.x)dx, a^= I K^{x x)dx, . . 



00 OO — 5C 



Ku{x,a) = fK{x,y) Kn-i{y,a) dy'{,i =1,2.3 . . ) 



and 



K,{x.ct) = K{x.a) 



From K{xy), wliich may be written 



1) Monalshefte f. Math, und Phys. 1904 p 121. 



