# Estimate error in approximating integral with sum

In a paper, for practical purposes the following integral is approximated as a sum, $$\frac{1}{2\pi} \int_0^{2\pi} f(\theta) e^{-il\theta}d \theta \approx \frac{1}{M}\sum_{v=1}^{M}f(v\Delta\theta)e^{-ilv\Delta\theta}$$ It is given an expression for the estimated error $$err= -\frac{2 \pi^3}{3M^2} \frac{d^2 f(\theta) e^{-il\theta}}{d\theta^2} \sim -\frac{2 \pi^3 l^2}{3M^2}$$ but no references are given on how the expression is found. Any ideas? Have you seen it elsewhere?

For completeness, here it is the paper (link, Eqs. 23-24), even though it is not really usefull to read it.

• Did you have a look at the errors for Riemann sums? – Clement C. Apr 27 '15 at 2:13
• It must be that formula, I wasn't aware of these methods, thanks. – Nicola Apr 27 '15 at 2:27