( 203 ) 



Numerical evaluation of the iiitofj,i-al for iiol loo larjï^c values of .v 

 offers no (lifriculties, as the series /> {.r,iii) and AI{iV,in) converge 

 rapidly. Because of the equation 



1 / — /( — 111 — 



r{m)J (,/•, m) = ^ \/ji I e ''" a 2 ,/,j 







the result will alwaxs l)e a posilive nuinl)ei" and llic integral will 

 not A'anish for any real \alue of /'. 



A few further remarks may be made. Firstly we may state, that 

 /{■r,iN) is intimately connected with Bkssel's function /„ (.r). In fact, 

 by means of the usual expansion of ,7,, (,/) we may verify the relation 



f{.c,m)= - 







J _1 



ft' 



From this we infer, that for positive integer \alues of //< the origin 

 ,t' = ceases to be a singular point, and that ƒ(./;,///) can be expressed 

 in unite terms. We shall find by actual substitution of the Hnite 



expressions for J _^^.A xe - 1 and J ^^^ _ ^ [ xc - J 



/( ^ Hi 1 



/(.,., ,„) = i' . ^- f ^Y"-' V -•'-n±i): . riY: 



• ^ 2 (m— iyV2y ^^ kl(,a—\—li)! \%v) 



h = o 



However this result may be obtained in a simpler ^va3' as follows. 

 It can be shewn, that /'(.'', in) obeys the relation 



r(m) 



and since we have 



J< = — e * ■ , 



1 + r-^ 2 



we get for all positive values of in 



t (,r, m) = — . J) — - - , 



a result that can be identitied with llial obhiined before. 



The singularity in the origin ,t' ^ becomes logarithmic, when 

 2iu is an odd integer 2/-}- J. The e\|)ression of /'(«,;//) is in this 



14 



Proceedings Royal Acad. Amsterdam. Vol. VII. 



