14^ 



it «2 



thus finall}' 



ƒ4 sin xu y — r 



e du = Vjt I e-^' da , 



u 



1 2" . n! J 



14. Prof. RuNGE gives the solution of the integral equation 



CO 



f{n)-=JK{x)ip{u^x)dx (31) 



00 



where f{u) and /v(.r) are given fnneJions and (f{,r) is required, by 



means of Hkrmite's functions. 

 He assumes 



K{x) = .-- [ajl,{x) + ajl,{x) + ajl^{x) + ...] 

 ^ {x) = e-^' [i>JI,{.r) + bjl^ix) + /,,//,(.') + ...] 



whioli gives 



00 



f{u) = ^ a,n K Ce-^' [I„ (x) f-('^+^r' ƒ/„ (m + x) dx 



00 



or, after some reductions 



]/2 (1/^)'"+" 



If now, the given function /{u) is expanded in this form 







1/2" L" ' t/2' ' ' {v^r 

 we have from (31) 



and it is evident that from these relations the coefficients h maj be 

 determined. If /(?/) and (p (.v) were the given functions, the same 

 relations would be sufticient to determine the function /v(.t). 



15. The preceding reduction rests on the formula 



■i-Cn^^uiy)] (32) 



