Convergency of Fourier s Series. 143 



for these points the integrals 



i 



S 



'0 2 1 



are indeterminate in value. 



33. We may therefore summarize the conditions under 

 which Fourier's series is convergent as follows, taking first 

 the case where the function F has no infinite values— the 

 case of a function having infinite values being discussed 

 later. In order that the series 



1 ff 1 n rn 



77- F(u)diH Scosw.ri F(v) cos nv'fto 



llT J-Tr I* 1 J -ir 



H — 2 sin nx \ F{c) sin nv"dv 



7T 1 J~rr 



may he convergent when n = co for any value of x 

 (i.) The coefficients must be finite and determinate ; 

 (ii.) The nth coefficient must vanish when n— 00 . 

 These are conditions that hold in the case of every series, 

 independently of its particular character. They are therefore 

 necessary conditions, but they are not sufficient. 



34. The first condition is satisfied if the function which 

 determines the coefficients is not indeterminate or discon- 

 tinuous over a finite range of values of the variable, but is 

 continuous and determinate except, possibly, in the neigh- 

 bourhood of a finite number of points where it may have any 

 number whatever of discontinuous, indeterminate, or singular 

 values. The second of the above conditions is also fulfilled 

 under the same circumstances. For, if we take the coeffi- 

 cients 



1 C 1 * if* 



— I F(t;) cosnv~dv, — I F(i>) sin mi'dv, 



and divide the variable all through by n, we get 



— F(-)cosi?cK — F(-)sinudv. 



W7rj_ njr \ni mrj- nir \nj 



Then breaking up each integral into n elements of range 2-7T 

 and applying the method of (14) we can show that the inte- 

 grals vanish when n is infinitely great. 



35. The condition given above to ensure that the coefficients 

 of the series are finite and determinate (namely, that F(v) 

 must be determinate and continuous, except in the neighbour- 

 hood of a finite number of points) is a special case of 

 Biemann's general condition as to the integrability of a 



M 2 



