FOURIER'S SERIES 



387 



Example. Expand the function t/ tf as a Fourier's Series, 

 working between the limits x - n and x - TC. 



Then y = B + Bj cos # + B a cos 2# + . . . B n cos nx+ . . . 

 Aj sin x + A 2 sin 2# -f . . . ^ sin no: + . . . 



and 27tB = y dx 



J 



f* 

 J-, 



also 



-L 

 -L 



y cos nx dx 

 e" cos nx dx 



in nx 



- e 6 cos nx + ne x sin 

 n- + 1 L 



-3 -{e* cos nrc e~ cos ( 



n -\- l 



cos mr, 



and 



cos nTt 



_ 

 When n is odd, cos WTC 1 and B n = -- - 



and B 1= - 



^ 



, B 3 = -, B 5 = - 



267C 



When n is even, cos nn = 1 and B n = 

 and B 2 - r . 



also 



7t(n 2 



57C 177C 



sin nx dx 



TtA,, = y sir 



= 1 et* sin nx dx 



J w 



-s 7 e* sin rw; ne* cos 

 n^+ 1 L 



J_. 



