Inspired by this post, I am trying to do a derivation of a Poisson summation formula.
My starting point is this: $$ \frac{1}{2\pi} \int^{\infty}_{-\infty} e^{i k x} dx=\delta(k) $$
I simply wish to take the continuum by the following, with $k_n=\frac{2\pi n}{L}$: $$ \sum_{n \in \mathbb{Z}} e^{-in (x_1-x_2) \frac{2\pi}{L} } =\sum_{n \in \mathbb{Z}} e^{-i k_n (x_1-x_2) } (k_{n+1}-k_n)\frac{L}{2\pi} \to \int e^{-ik (x_1-x_2)} dk \frac{L}{2\pi} =L \delta(x_1-x_2) ---Eq.(1) $$
The arrow $\to$ is taking the continuum, so:
$$ (k_{n+1}-k_n) \to dk,\\ \sum_{n \in \mathbb{Z}} \to \int $$
Follow the answer of this post on Poisson_summation_formula, I should have
$$ \sum_{n \in \mathbb{Z}} e^{-in (x_1-x_2) \frac{2\pi}{L} } =\sum_{m \in \mathbb{Z}} L \delta(x_1-x_2-mL) ---Eq.(2) $$
Question 1: so what is wrong with my derivation on Eq.(1)?
Question 2: how to derive Eq.(2) by taking the continuum?
Thank you.