Fourier transform on a loaded string

[Mrbluesky323]

Homework Statement

Can someone tell me how to Fourier transform this quantity:
$$\Sigma$$ (x_(j+1) - x_j)^2

where the sum is from j=1 to N

Homework Equations

Define the Fourier transform as
x_j = $$\Sigma$$ A_k *exp(-iqkj)
**Where i is sqrt(-1)
**The Sum is from k=0 to (N-1)
**q = (2*pi)/N
**Assume periodic boundary conditions (i.e. x_(N+1) = x_1)

The Attempt at a Solution

I get to

$$\Sigma$$_j ($$\Sigma$$_k A_k* exp(-iqkj) (exp(-iqk) - 1))^2

I'm told the answer one arrives at ought to be:

$$\Sigma$$_k A_k* A_-k*(2sin(qk/2))^2
I'm not sure how to get there from where i am.
Any help would be appreciated.

 

[BigJDubs]

Solution: We can expand the equation as follows: \Sigmaj (\Sigmak Ak* exp(-iqkj) (exp(-iqk) - 1))^2 = \Sigmaj (\Sigmak Ak^2* exp(-2iqkj) - Ak* exp(-iqkj) - Ak* exp(-iqkj) + 1)^2 = \Sigmaj (\Sigmak Ak^2* exp(-2iqkj) - 2Ak* exp(-iqkj) + 1)^2 Now, since we have periodic boundary conditions, we can rewrite x_(N+1) = x_1 as: exp(-iqk(j+1)) = exp(-iqkj) * exp(-iqk) Substituting this into the equation above gives us: \Sigmaj (\Sigmak Ak^2* exp(-2iqkj) - 2Ak* exp(-2iqkj) * exp(-iqk) + 1)^2 = \Sigmaj (\Sigmak Ak^2* exp(-2iqkj) - 2Ak* exp(-iqkj) (exp(-iqk) - 1))^2 Now, we can use the fact that sin(qk/2) = (exp(-iqk) - 1)/2i to rewrite the equation as: \Sigmaj (\Sigmak Ak^2* exp(-2iqkj) - Ak* exp(-iqkj) * (2sin(qk/2))^2)^2 = \Sigmak Ak* A-k*(2sin(qk/2))^2