I am solving an initial-boundary value problem and I got to a point that I have to calculate the coefficients knowing that:
x=Σ{An sin(((2n+1) π x)/(2L))}


One way to calculate the coefficient is the Fourier series, isn't it?


So I thought the following:


We expand the function f(x)=x in an odd way at [-L,L].


So we can write the function as a Fourier series.


Can we write it as followed?


f(x)=Σ{bn sin((2(2n+1) π x)/(2 * 2L))},
where bn= 2/(4L) ∫_{-L}^L{f(x) sin((2 (2n+1) π x)/(4L))}dx


Or can we not do this in this way?