Convergency of Fourier s Series. 139 



functions, however, it is necessary to consider the cases in 

 which this condition does not hold. 



28. The investigation of Dirichlet involves the first and 

 second of these conditions, but not the third. The third is 

 replaced by the more general one that F(^) must not have an 

 infinite number of maxima and minima between +7r. In 

 Dirichlet's investigation this condition is applied to the 

 function throughout the whole extent of the integral S„, that 

 is for all the values of the variable of integration 6. This, 

 however, is not necessary. For it has already been shown 

 that the portions A and of the integral vanish when n = x> 

 if only the function is finite and continuous — the nature of 

 the continuity being immaterial. The third condition should 

 therefore apply only to the infinitely small range of values of 

 "F(0 + #) which lie on either side of 0=0. We shall now 

 show that this condition is sufficient to ensure that the 

 integral 



j: 



V(g+*)-r(.»)] * n(2 " +1) * g ag 



-h 2 U 



vanishes when n = co , and that therefore Soo=F(#). 

 This integral can be put into the form 



J. 



♦ 1^3«, 



h being infinitely small, while m is infinitely great and (f)(0) 

 infinitely small between and h. Since (f)(6) has not an 

 infinite number of maxima and minima, it will ultimately 

 preserve the same sign, and either constantly increase or 

 constantly diminish between and h. Let it constantly de- 

 crease. Then, dividing the variable by m, we get 



C mh . /0\sin 



J. +U-9 



This integral can now be broken up into the sum of a series 

 of elements which are alternately positive and negative and 



constantly diminishing numerically (since — -x— and <p(— ) 



diminish numerically). Hence the integral becomes an alter- 

 nating series with constantly diminishing terms, and its value 

 is therefore less than the first term, which is itself infinitely 

 small. That is, the integral vanishes. Again, let (f)(6) con- 

 stantly increase between and h. Then its greatest value 

 will be <f>(h), and \_<f>(h) — (f)(6)] will therefore constantly 



