'831 



so that ill lliis case it may be concluded from the integTal formula that 



I ƒ ( e T j cos 2jit:v d.v = h ' — ƒ ( g~ T J . (« = iw — 1) . . (H) 







As may be proved the equation (II) remains true if we suppose 

 t=zO, and if the expansion in series 



is made use of, we get in this exceptional case 



:£ = — /(1)= -~ 2 i~]h. {a=:4:W—l) 



m = \ \ (-t J tn \/a' ay « h=\ \a/ 



The results found by Stiklt.irs have been derived with this, the 

 equations (I) and (II) may now, however, be used, to find other 

 results less known in the theory of numbers. 



B'or real values of (ü the function ƒ (^'-='') has the properly of approach- 

 ing rapidly to zero for positive and negative values of .i' of increasing 

 modulus. This leads to the conclusion that Fourier's general sum- 

 mation-formula 



" v"' F{^-j-7i) = if (,v) cbj + 2"^" (f(,v) cos 2jin (y-^) dy 



— 1 



may be applied, if we write 



•2nx ' 



F{.v)=f\e'T), 



and if we suppose ^ 5 <^ 1. 



Distinguishing again the cases a = 4,}i)-\-\ and (7 = 4?/' — 1, the 

 value of the integrals in the righ(-hand member may be determined 

 by means of the equations (I) and (II). It should be taken into 

 consideration in the summations in the left-hand member, Ümt f (e~''-^) 

 changes its sign together uith ,r or not, according as a is eqnal to 

 4:1V + 1 or to 4:W — I . 



In this way the two following general equations are deiived from 

 the summation formula. 



, (III) 



= 2 I— :E sill -Jitiifl e V ) , (rt = iw + 1) ) 



7 .=1 V J 



55 



Proceedings Royal Acad. Amsterdam. Vol. XVII. 



