Then 



-7rA n =- I y si 

 2 Jo 



FOURIER'S SERIES 



sin n0 dQ 



889 



cos wO 20 sin w0 2 cos n8 



c 2 f 6 2 co 



5 



7C" L n 



c 2 / 7i 2 cos ms 2 \ 



31 --- h , (cos nn- l)t 

 7t 2 ^ n n 3 v 'J 



J 



2C 2 / 7C 2 



and A,, = - \ 



7t 3 v 



n 



2 . ,. 



f (cos nit -1) 

 3 ' 



2 C 2 /^a 4 \ 



When n is odd. cos me = 1 and A_ = - { --- - ) 



TC J \n n 3 / 



2c 2 . 2c 2 /u 2 4\ 2c 2 /7r 2 4N 



and Al - _(.' - 4), A 3 - ^ (- - -j, A. ^ (- - ), etc. 



2c 2 7C 2 



When n is even, cos n?r = 1 and A.. = -- s- 



7c 3 n 



, 



d 



7T" . 



and y = ?L\ (u 2 - 4) sin - ^-sin 20 + (~ - -) sin 30 



Since = 

 c 



2c 2 



37ra? 







- sin 



4 



+ . . 



192. Harmonic Analysis. If the graph of a periodic function is 

 given, it is possible to analyse the curve and express the result 

 as a Fourier's Series. For if y is any ordinate of the curve and 

 the base line of the curve is made to extend from to 27T for 

 a complete period 



Then 



y = /(0) 



! cos + B 2 cos 20 + . . . B w cos n0 + 

 j sin + A a sin 20 + . . . A,, sin w0 + 



