# Fourier Series For Function Not Centred at Zero

1. Nov 4, 2014

### modulus

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.

Last edited: Nov 4, 2014
2. Nov 4, 2014

### LCKurtz

It would be good if you gave us more equations and less prose so we understand better what is really going on.