Sokhotsky-Weierstrass theorem

$\lim_{\epsilon\to 0^+} \int\!dx\, \frac{1}{x \pm i \epsilon}f(x) = \mathcal{P} \int\!dx\, \frac{1}{x} f(x) \mp i \pi f(0)\,$ ,

where $\mathcal{P}\,$ denotes the Cauchy principal value of the integral.

Distribution

$\lim_{\epsilon \to 0^+}\frac{1}{x \pm i \epsilon} = \mathcal{P} \frac{1}{x} \mp i\pi \delta(x)\,$ .

The result generalizes as follows:

$\lim_{\epsilon \to 0^+}\frac{1}{(x + i \epsilon)^{n+1}} = \mathcal{P} \frac{1}{x^{n+1}} - i\pi \frac{(-1)^n}{n!}\delta^{(n)}(x)\,$ .

↑ A. Capri (2003). Problems & Solutions in Nonrelativistic Quantum Mechanics. World Scientific Publishing, 116. ISBN 978-9810246501.