384 



c 4c f 1 1 1 



Then y = -A cos x + - cos 3x -f cos 5a: + cos 7x + ... V 



2 TiH 9 25 49 J 



working in terms of c. 



1 4m f 1 1 ^ 



or y = -mTC { cos x + - cos 3# + cos 5x + . . .} 



2 7T v. 9 *sO J 



working in terms of m. 



Example. Expand the function y = mx 2 as a cosine series, 

 knowing that y = c when x = TE. 



Then m = , and y = 5 # 2 



and ?/ = B + B 1 cos a: + B 2 cos 2a? + . . . B n cos nx + . . . 



pr pr 



Then B I dc = I y dx 



Jo Jo 



c f- 

 7iB = gl * afc 



" Jo 



C 7C 3 , ^ C 7t 2 



' = 7? 2 



Also I ?/ cos nx dx == B n l cos 2 nx dx= 7tB r 

 Jo Jo 



2 



But I i/ cos nx dx = I 2 



Jo u Jo 



cos 



c Hr 2 sin na; 2a; cos nx 2 sin nafl* 

 ^^__^^__ i ^^^^^^__ _^^___ 



^O I O Q 



TirL n w^ n 8 J 



c 2-n 



= x z COS W7C 



7t 2 n 2 



c 4 

 Then B n = 5 cos me 



TC /C- 



c 4j 

 When n is even, cos me = 1, and B n = r -r 



71 II B 



-5 ,, 



B 2 = - 2 -, B 4 =--, B 6 = _-,e.tc. 



c 4* 

 When n is odd, cos nit = 1, and B n = -- ^ 5 



TC /i 



C4.,, c4_ c4 



and 6,= ---, B,-- a53 , B 5 - -3 ^ etc. 



