Integrals of functions with poles on the real axis (Sokhotski–Plemelj theorem) in the sense of Distributions/Generalized Functions.

Question

The Sokhotski–Plemelj theorem [Wikipedia link] states that $$ \lim_{\epsilon \to 0 } \int_{\mathbb R} dz \; \frac{f(z)}{z \pm i \epsilon} = \mp i \pi f(0) + \mathcal{P} \int dz \; \frac{f(z)}{z} \;, $$ for $f$ analytic on $\mathbb C$. This is often expressed in physics literature as $$ \frac{1}{z \pm i \epsilon} = \mp i \pi \delta(x) + \mathcal{P}\frac{1}{x} \;, $$ which makes heuristic sense. Is this rigorously valid in the sense of distributions, i.e given any test function $\varphi \in \mathcal{C}_c^\infty(\mathbb R)$ (smooth and compactly supported), can we show that $$ \int_{C^\pm} dx \; \frac{\varphi(x)}{x} = \mp i \pi \varphi(0) + \mathcal{P} \int dx \; \frac{\varphi(x)}{x} \;, $$ where $C^\pm$ is the real line indented around the origin into the upper and lower half planes respectively, and suitably closed at infinity. In attempting to mimic the proof for analytic functions, I fail to make sense of the right hand side contour integral - how to have suitable test functions on the complex plane that are analytic (so that we can use Cauchy's theorem), as anlaytic functions that vanish on a segment of real axis will necessarily vanish everywhere.