Suppose I know that a function is analytic in the upper half plane and that for real $x$ we have $\lim_{\epsilon \to 0^+} \operatorname{Im} f(x+ i \epsilon) = \delta(x)$. Can I conclude that $f(z) = \frac{-1}{\pi z}$? Is the Sokhotski-Plemelj formula unique in this sense?
(Context: I am reading a physics text on scattering theory and they appear to me to be using uniqueness in this sense.)