Convergency of Fourier's Series. 141 



vergent, and tend to the limit F(ar), even when F(0 + #) has 

 an infinite number of maxima and minima, provided that at 

 all the points where the function oscillates, the numerical 

 value of F(# -f ± 8) — F (x + 0) is always less than BB a , when 

 8 tends towards the value zero, B being a finite constant, and 

 a sl positive exponent. Here again it is really necessary to 

 apply the condition only to the infinitely small range of 

 values of the variable of integration which lie on either side 

 of = 0; for if the condition is satisfied for these values, the 

 integral 



£ 



LF^ + ^-FW] ™^ 1 ^ 



±0 



vanishes, and therefore, as before, S 00 =F(a?). This integral 

 vanishes under the given conditions because its value cannot 

 be greater than the value it would have if sin (2n + 1)^0 

 were replaced by unity, and all the negative values of 

 ~F(x-\-0)— F(#) made positive. Hence, since for all values 

 of between and h, h being infinitely small, the numerical 

 value of ~F(%±0) — F(#) is <B6 a , the integral cannot be 

 greater than 



■J.' 



2B 0^0, or 



2BA a 



which is infinitely small, since a is & finite exponent. Thus, 

 the function may have an infinite number of maxima and 

 minima of this type, and still give rise to a convergent 

 Fourier series, whose converging limit is F(#). 



31. It is not necessary that F(0-t-<2?) should be finite 

 throughout between +7r. It may become infinite at a finite 

 number of points a x a 2 . . . provided that 



Iim f«+M2< 

 — q F(0+*)30 



J a— ^e 



vanishes, /j, t and /^ being any independent positive fractions. 

 For if this vanishes, then 



lim f a + T(fl + g) e 

 6=0J a _^ t sin# 



also vanishes, unless passes through the value zero, for it 

 tends to the value 



1 hlim C a +£ 3 ' ^ 1 

 sinaU = 0J a _ Mi< v '^ J 



Phil. Mag. S. 5. Vol. 42. No. 255\ Aug. 1896. M 



