I therefore need to find the odd extension function for:

$ phi (x) = begin {case} x & text {,} 0 leq x leq 1 \ 1 & text {,} 1 leq x leq 2 \ 3-x & text {,} 2 leq x leq 3 end {boxes} $

I will use this: $ f_ {odd} (x) = begin {case}

-f (-x) &, -3 leq x <0 \

f (x) &, 0 leq x leq 3

end {cases} $

So, where I need help, it's more here:

For the Fourier series, I can not find it, even with the help of the Fourier Sine series that I have to use to solve this limit value problem:

$ begin {cases} u_ {tt} = c ^ {2} u_ {xx} &, t> 0 ;, x ; epsilon ; (1,3) \ u (0, t) = u (3, t) = 0 & \ u (x, 0) = phi (x) &, x ; epsilon ;[0,3] \ u_ {t} (x, 0) = 0 & end {cases} $

I really do not know how to solve the last problem. If anyone could help me, thanks in advance.