195 



«00 00 



e-t Jt — _ e-'/-{e-2'//jj(Ay) 4-e23//ï(e-2//)|,/y e—2>/"du S (M-u') 



J V^J 'J o n.' 



o 



Now is ^) : 



00 .>;"' 



O 'nf 



in wliicli e/5 represents the function of Bessel of order zero. From 

 this it ensues consecinently that: 



^X 00 



1 = — — - iy-^-'^ini, {tV/)-\-emi{e-^)\di/ (e-^->/"J^{2im/yiu. 

 u 



As is known, 



J„ {2iuy) = ^ 



« |42»iy2n 







u=o {niy 



so 



ƒ* 00 w2h /"» X 1 /"• 



f-2«-/ J {2iuy) du = :E ±^ \ e-^-uyifi»du= 2 ( 







Introducing this we have 



•4 r°^ X (2n)' 





 or 



. .2n^> r ~ "= ~ i^^j ^'~'' ^''^ ^^'"^ "^ ''" ^^' ^^^"'"^^ ^^ * ^^^^ 



According to an integral used before, is 



00 00 00 00 



1 [e-mi^ {ê!i) -1- e^yli{e-'^y)] dy= — 2\t cos tdt i — ^— = - Hm i cos t arc tg — dt. 

 ' 



Formuki (13) may also be written as follows: 



1 4 ^. r^ 2y 



— iim I cos t arc tg — dt . . (14) 



» f2n\ 1 I/: 



2n\ 1 

 /I J22M^ 



^=0 



By multiplying formula (11) by ~ and by summalioii from 7i =0 



11 .' 



to y^ = X we find : 



1) These Proceedings XV, p 1246 (9). 



