Convergency of Fourier s Series. 133 



Hence in such a case S. = i [F(>- 0) -f F(* + 0)] *. F (0 + *) 

 may have such discontinuities for other values of 6 as well, 

 provided their number is finite. For if we break up the 

 integrals A and C between neighbouring discontinuities into 

 separate portions, we may show, as in (14) and (15), that each 

 of these portions vanishes when n = <x> . Hence, since there is 

 only a finite number of them, their sum vanishes, and there- 

 fore A and C vanish when n = go ; so that, as before, the 

 value of Soo depends only upon the value of the infinitely thin 

 strip which lies between +h. Consequently F(0 + #) may 

 have any finite number of discontinuities between + it, the 

 value of So, at any discontinuity being the mean of the values 

 to which Y(6 + x) tends as the discontinuity is approached 

 from either side. 



18. If F(6 + x) is not periodic, we may regard the portion 

 of it included between +7T as a wave of an arbitrary periodic 

 function with, in general, finite discontinuities at +7r, +37t, 

 &c; so that when x= +tt, Sa, = J[F( — 7t) + F(7t)] by (17)f. 



* Or thus, 



2* LJ_f ( + * } P d + J ( + ° W d J 



Jo 2# 



Hence, applying to this the method of (16), we get 

 |[F(*-0)+F(*+0)]. 

 t Or thus :— If F(6+x) is not periodic, 



If x lies between and ir, 



and if x lies between and ~ n, 



In both cases the function under the sign of integration becomes infinite 

 only when 0=0, and the integration can therefore be effected by the 

 methods given above. 



Putting x=tt in the former, or x— — -n in the latter, we get 



the limit of which, when w = oo , is 



S..i[F(-ir)+F(ir)]. 



