829 

 Mathematics. — ''On on integral formula o/' Stieltjes." By Prof. 



J. C. KlXVVER. 



(Cominunicaled in the meeting October 3i, 1914). 



Ill the Proeeediiigs, and Comiminieations, Physical Section, series 

 3, 2. 1886, |). 2 JO, Stieltjes treats of detiiiite integrals, referring- 

 to the function 



.t (//) = — , — - — = ^ r- 



I— If' „=i\aj 



In this ftmction a stands for a positive odd integer without qua- 

 dratic factors, and (- lepresents LegendiïI'/s synd)ol with the ex- 

 tension given to it by Jacobi. 



As poles of tlie function /'(//) only the points i/ ^ e " are to be 

 taken into consideration, and for the residue, belonging to such a 

 pole, one finds 



1 " 2:: i \ e 



a fi=\ \a 



Fioni the well-known fundamental equation 





/■-?)■ (3 .. 



-ik 



it follows, that a pole is only to be found in those points y=r 6^ « , 

 in which / is jtrinie to a. Consequently // = 1 is not a pole of the 

 function, and we have. 



1 /'=^'-i rh\ 



ƒ(!) = -- V __ U 



a /,=i \a J 

 from which it follows that — '(/'(I) is ecpial to the sum of the 



jiumtters smaller than a, for which ( J=-j-l (residues), diminished 



with the sum of the numbers smaller than a, for which ( )= — i 



i^ (non -residues). 



In the paper cpioted, Stiicltjes considers the definite integrals 



0(3 OO 



J' ,, ^ . dt'^ , , C atx 



J (^~') -s"* 2^ ^•'' •'"'^ I J {^-'') cos — chv, 



