I am reading up on the Sokhotski-Plemelj theorem, and so far I've seen it being applied on equation with the general form (ref: http://scipp.ucsc.edu/~haber/ph214/Plemelj.pdf):
$$\lim_{\epsilon\to0}\frac{1}{x-x_0\pm i\epsilon} = P\frac{1}{x-x_0} \mp i\pi\delta(x-x_0)$$
Can the same theorem be applied to the following equation with an additional sqrt term? $$\lim_{\epsilon\to0}\frac{1}{\sqrt{x-x_0\pm i\epsilon}}$$
I can only think of bringing out the sqrt term and applying the formula to get the same result but with the additional sqrt term:
$$\sqrt{\lim_{\epsilon\to0}\frac{1}{x-x_0\pm i\epsilon}} = \sqrt{P\frac{1}{x-x_0} \mp i\pi\delta(x-x_0)}$$
Is there any way to simplify that? Or is there another way of applying the theorem? Thank you.
