Fourier Series For Function Not Centred at Zero

[modulus]

Homework Statement

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).

Homework Equations

The domain for the spatial part was [0, L].
Given spatial function at t=0 (boundary condition). Call it Uo(x).

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.

 

[LCKurtz]

Homework Statement

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).

Homework Equations

The domain for the spatial part was [0, L].
Given spatial function at t=0 (boundary condition). Call it Uo(x).

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.

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