FOURIER'S SERIES 885 



Therefore 



C 7C 2 4cf 1 1 1 1 



w -= 3 -- ;{ cos x - cos 2x 4- cos 3x - cos 4# + . . . } 

 7C 2 8 TiH 4 9 16 J 



c 4cf 1 1 1 1 



-- r,1 cos x - cos 2,r -\- - cos 3# - cos 4r + . . . \ 

 3 TC-V 4 16 J 



working in terms of c 



m? fill 



r y = q 4w| cos # ~ T cos 2 * + Q cos ^ ~ Tft cos ^ 



working in terms of m. 

 190. Fourier's Series. The general form of Fourier's Series is 



y = /(#) = B + ^i cos x + ^2 cos 2tf + . . . B n cos na; + . . . 



+ Aj sin x + A 2 sin 2x + . . . A^ sin nx + . . . 



Since the integrals I sin nx dx and I sin nx cos mx dx do not vanish 



when taken between the limits and TC, but they do vanish 

 when taken between the limits and 2n, or between the limits 

 - TC and TT. Fourier's Series can only be worked when the 

 necessary integration is performed between the limits and 2it, 

 or between the limits n and TC. 



In working with this series, three operations are necessary, 

 the first to find the initial term B , the second to find the general 

 coefficient B n , and the third to find the general coefficient A^ 



(1) If we integrate throughout with respect to x between the 

 limits and 2;c. 



Then 



- B 



I yfa 



Jo 



f2w P2v Kw PZw 



I C& + B!! cosrrda:+B 2 i cos2#d!#+. . . B n | cosnxdx+. . . 



Jo Jo Jo Jo 



rfZw fZ* 



sinxdx + A. 2 \ sin 2xdx+. . . AJ sin nx dx + . . . 



K| Jo Jo 



and all the integrals on the right-hand side vanish except B 1 dx, 



Jo 

 which becomes 27tB . 



Hence 27rB = I y dx 



Jo 



and B - -, dr 



2 B 



