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We want to calculate the $\lim_{\epsilon \to 0} \int_{-\infty}^{\infty} \frac{f(x)}{x^2 + \epsilon^2} dx $ for a function $f(x)$ such that $f(0)=0$. We are physicist, so the function $f(x)$ is smooth enough!. After severals trials, we have not been able to calculate it except numerically. It looks like the normal Lorentzian which tends to the dirac function, but a $\epsilon$ is missing.

We wonder if this integral can be written in a simple form as function of $f(0)$ or its derivatives $f^{(n)}(0)$ in 0.

Thank you very much.

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    *We are physicist* sounds like ""We are Borg. Resistance is futile..."2012-07-17
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    We may have a problem at $0$: for example, if $\chi$ is a bump function (smooth, with compact support, non-negative, and constant equal to $1$ in a neighborhood of $0$) and $f(x)=x\chi(x)$, the limit is infinite. But it's finite if we add the condition $f'(0)=0$, as Taylor's formula shows.2012-07-17
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    Physicists (I am myself one) tend to replace this integral with $ (\pi/\epsilon) f (0) $ (letting $\epsilon$ not going all the way to 0 ;-). However, one needs more than smoothness to justify this.2012-07-17
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    Has the function $f$ poles in the complex plane? Is it analytic? For an entire function the result would be $\frac{\pi f(i\epsilon)}{\epsilon}$ (well if the integral on the circular contour in the half plane goes to $0$ of course).2012-07-17

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