Convergency of Fourier s Series. 147 



i. The function must not be indeterminate or discontinuous 

 over a finite range of values of the variable ; 

 ii. The integrals 



i 



[F(.r + g)-F (; ,)] sin(2 " + 1) ^ 9g 



must vanish, or tend to definite limiting values, when w = qo 

 and h is infinitely small. 



89. This last condition is somewhat analogous to the con- 

 dition which holds in the case of Taylor's Theorem when applied 

 to numerical functions, namely, that the u remainder " after the 

 nth term must vanish when n is increased without limit ; — a 

 sort of test to be applied to each individual function dealt 

 with, because we have no means of determining in a general 

 manner when the condition is fulfilled. If the first condition 

 is satisfied, the coefficients of the series are finite and deter- 

 minate, the nth. coefficient vanishes when n — <x> , and the value 

 of the series at any point x depends only upon the infinitely 

 small portion of the function F which lies on either side of 

 that point. If the second condition is satisfied, the series is 

 convergent, and if, further, the integral involved in this con- 

 dition vanishes, the converging limit of the series is F(x). If 

 the first condition is not satisfied, the coefficients of the series 

 are indeterminate and meaningless, and the series cannot 

 therefore be formed. Whether the function can still be 

 represented by an harmonic series in such a case — the coeffi- 

 cients being determined otherwise — is a matter with which 

 we are not now concerned, nor are we concerned with deter- 

 mining whether the same function can be expanded harmoni- 

 cally in more ways than one. We are concerned only with 

 determining the most general conditions under which Fourier's 

 method of expanding functions into harmonic series is appli- 

 cable. In cases where it fails, we have no general method of 

 proceeding. 



40. If the function has infinite values, two cases may arise 

 according as the function has or has not an infinite number 

 of maxima and minima where it is infinite. In the former 

 case, as shown above, the series is convergent (except, of 

 course, at the points where the function is infinite) pro- 

 vided the function becomes infinite only at a finite number 

 of points, and that its integral vanishes when taken between 

 limits infinitely near to and on either side of each of these 

 points. In the latter case, for example in the case of 



- cos -, where - is infinite when x = 0, and cos— has an 

 x x x x 



