Question

(a) Find the Fourier series for f(x) = -x, -1<x<1 f(x+2) = f(x)

Answer 1

(8) - - - しくてく」 T: 2L = 2 > L 깃 어 The constant term is one half of Ao L Q. -- f(u) cos Man L dx -L (~) cos O an de L 11 - [ ] [-] ) SE9 1 11 1 Q. = 0 The rest of cosine coefficient 세) cos nan dx L 쨷 an : ) . -L - - ic) (-n) cos nan dx -

an= sin nan Пл - ani 1 a sin nan + na n²x² al - Sinne di this case max] - ( Men han cons] I sin (ona) son T (x4-)sor it an: I sin na + I cosna n²t² zu ' I - I ano-1 cos na 0 + O + 쁜 na² The sine coefficients area - bn. ſ Fla) sin norma de L -L bn (~) sin nan nan dx 1.x S --:- + monde] bn cos -1 Na sin nix x]! + 24 cos nan bn n²A² NK - (1) vu uis (- na cos (ana) + L sintna) 1 Cosna bn=1 z X ₂U si sin(n) nza? tyusos cos na to F (6 + ram ") bra-1 nt 2 Cos n bn nã

2 (1)" 2 bra 2 ; n = odd 11 nx na 2 na even ht fo E = an cos han + a ao 2 + han bn sish nau L - stan ] MA Since ao = anao f(n)- Ž (-1)" 2. sin nan nal n f(x) 2 E (1)". sin han k THIS IS THE REQUIRED FOURIER SERIES