so I need to find the odd extension function for :
$\phi (x) = \begin{cases}x& \text{ , } 0 \leq x\leq 1 \\ 1& \text{ , } 1 \leq x\leq 2\\ 3-x& \text{ , } 2\leq x\leq 3 \end{cases}$
I'll use this : $f_{odd}(x)=\begin{cases} -f(-x)& , -3\leq x <0\\ f(x)&, 0\leq x \leq 3 \end{cases}$
So , where I need help is more here:
For the Fourier serie , I haven't be able to find it even using the Fourier Sine Series that I have to use in this limit value problem :
$\begin{cases}u_{tt} =c^{2}u_{xx}& , t>0\;, x\;\epsilon\; (0,3) \\ u(0,t)=u(3,t)=0 & \\ u(x,0)=\phi (x)& ,x\;\epsilon\;[0,3] \\ u_{t}(x,0)=0 & \ \end{cases}$
I really don't know at all how to do the last problem .. If anyone could help , thanks in advance.