Wikipedia gives the follow statement of the general Sokhotski-Plemelj theorem:
Let $C$ be a smooth closed simple curve in the plane, and $\varphi$ a complex-analytic function on $C$. Define $$ \phi_i(z) := \frac{1}{2 \pi i} \oint_C \frac{\varphi(\zeta)}{\zeta - z} d\zeta $$ for $z$ inside $C$, and $\phi_e(z)$ similarly for $z$ outside of $C$. Then for a point $z \in C$, $$\lim_{w \to z} \phi_i(w) = \frac{1}{2 \pi i} \mathcal{P} \oint_C \frac{\varphi(\zeta)}{\zeta - z} d\zeta + \frac{1}{2} \phi(z)$$ and $$\lim_{w \to z} \phi_e(w) = \frac{1}{2 \pi i} \mathcal{P} \oint_C \frac{\varphi(\zeta)}{\zeta - z} d\zeta - \frac{1}{2} \phi(z),$$ where $\mathcal{P}$ denotes the Cauchy principal value of an improper integral.
The article also states a "version for the real line":
Let $f$ be a complex-valued function which is defined and continuous on the real line, and let $a$ and $b$ be real constants with $a < 0 < b$. Then $$\lim_{\epsilon \to 0^+} \int_a^b \frac{f(x)}{x \pm i \epsilon} dx = \mp i \pi f(0) + \mathcal{P} \int_a^b \frac{f(x)}{x} dx.$$
I'm not entirely clear on the exact relation between these two theorems. Can either version be easily derived from the other? Are they both special cases of a more general theorem?
They're clearly related; they both deal with approaching a contour of integration with a similar integrand, with a jump discontinuity across the contour. But there are also some important differences; e.g. in once case the contour is closed, but in the other case it isn't, and in one case the function itself is changing inside the limit. So it isn't obvious to me how to derive either version from the other.