FUNCTIONS IN THE THEORY OF INTEGRAL EQUATIONS. 155 



which may he written 



(44) 



where a is any positive number ( 10). It is here supposed that f(s) is defined 

 outside ( TT, IT) in such a way that it has a Lebesgue integral in any finite interval. 

 In what follows we shall secure this by the rule 



as in the theory of FOURIKR'S series. 



Again, when 's lies in the open interval ( TT, 0), we have x = \s\, \s\ being a 

 point of the open interval (0, IT). Proceeding as before, we now find that the limits 

 of indeterminacy of (43) are identical with those of 



~ f' [/(<)+/(-W-^- f [/W-/l-0]<rt + -- L n|.| f" 

 TT .'n JTT Jo TT Jo 



+ - cos n | s | f" [/(<) +/(-<)] cos nt dt-...\ 



TT .'a 



and hence with those of 



- f f(-t)dt + - coslsl f /(-<) cos tdt + ... + - cos n\s\ \ f(-t) cosntdt + ... 

 ir Jo TT Jo IT Jo 



It follows that, when s lies in the open interval ( w, 0), 



p -. x Z JO 



aim 



whence it appeare that (44) is valid for these values of . 



Lastly, at either of the points IT, 0, IT, it is evident that the limits of indeter- 

 minacy of (43) are identical with those of 



+ ... + - cosn|s| [ %[f(t)+f(-t)]cosntdt+.... 

 Since 



Jo 2 J o 



( 15), we at once deduce that (44) is valid when s= 0. Recalling that /() is 



x 2 



