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.

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

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