References: continuum approximation of discrete sums?

[yucheng]

Is there more references for how accurate is the continuum approximation to discrete sums? Perhaps more mathematical.

What I've found:
https://lonitch.github.io/Sum-to-Int/
https://arxiv.org/pdf/2102.10941.pdf

Some examples are:
Sum to integral
$$\sum_{\mathbf{k}} \to 2 \left ( \frac{L}{2 \pi} \right ) \int d^3k$$

Density of oscillator modes etc

 

[Haborix]

Here is Terry Tao discussing the Euler-Maclaurin formula mentioned in your arxiv link: link.

EDIT: Also, is there a reason you posted this in the Quantum subforum?

 

[yucheng]

EDIT: Also, is there a reason you posted this in the Quantum subforum?

Thanks!

Yes it appears in many places in Quantum optics, so I was hoping that there is a less mathematically abstract analysis, especially that of the error in the approximation (for instance, applying it to a model physical system, comparing the exact sum vs continuum approximation), whether it causes deviations from experimental results.....

P.S. the zeta functions, bernoulli functions makes me want to cry, but if that's what's needed, then I'll have to slowly crawl my way there...

 

[yucheng]

I did find one! Serendipity!

Fermi's golden rule: its derivation and breakdown by an ideal model by J. M. Zhang, Y. Liu

Search in document:
in replacing the summation by an integral, the sampling step-length
https://arxiv.org/pdf/1604.06916.pdf