1. The problem statement, all variables and given/known data I was working on a problem where I had been given a differential equation to be solved using separation of variables. Two coordinates: a time coordinate and a single spatial coordinate (1-D problem). 2. Relevant equations The domain for the spatial part was [0, L]. Given spatial function at t=0 (boundary condition). Call it Uo(x). 3. The attempt at a solution I ended up with an exponential function in the time coordinate multiplied by an expansion that looked exactly like the Fourier Series of the spatial part. So, when I put t=0, I end up with Uo(x) on the left of the equality and what should be its Fourier Series on the right. Under usual circumstances, I could extend the function to the domain [-L, L] by simply assigning Uo(x) (resp. its negative) to the domain [-L,0] to get an even (resp. odd) function and a cosine (resp. sine) Fourier series. But the problem is that the right hand side has both cos and sin terms! So, how do I determine the coefficients of in my solution now? Is there any theory that can allow me to take a function centred at a point other than zero and shift it to get a Fourier Series for it in the relevant domain? From what I've tried out, I think it can be done, but it would involve making the arguments of the sines and cosines in the series2n*pi(x-a)/L where a is the point around which my function is centred. Edit: The spatial boundary conditions (other than the one mentioned above, that was a boundary condition for the time coordinate) force the Fourier Series spatial solution to retain both cosine and sine terms.