How to estimate the correction to the integral approximation of a discrete sum?

Question

In the following approximation,

$~\epsilon\sum_{n\in Z} F(\epsilon n)=\int_{-\infty}^{+\infty} dx~ F(x) + $ correction,

how can one estimate the `leading' order correction for small $\epsilon$?

Here the function, $F(x)$, and its all derivatives, $F^{(n)}(x)$, drop off rapidly at $\pm \infty$ such that the Euler–Maclaurin formula does not help. For example, $e^{-x^{2}}$.

score 1 · Answer 1 · answered Aug 09 '13 at 16:57

1

I would estimate the both the sum and the integral outside some compact by a small number (your function drops off on infinity quite fast) and then apply standard formulas on that compact for numerical integration using the trapeze/ladder approximations.

Did you try the method from wiki?

answered Aug 09 '13 at 16:57

TZakrevskiy

22,980

What I am looking for is an alaytic and closed expression for the correction as a function of $\epsilon$. – Jeong-Hyuck Park Aug 09 '13 at 18:22

How to estimate the correction to the integral approximation of a discrete sum?

1 Answers1