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I have a question about the spectral representation of an analytic function $G$ on a Riemann surface (specifically, the complex plane with a finite amount of cuts), i.e. the representation of the form $$ G(x) = \int dy \frac{\rho(y)}{x-y}, $$ where the density $\rho$ is defined by $$ \rho(x) = \frac{1}{2\pi i}\lim_{\epsilon \downarrow 0}\left(G(x-i\epsilon)-G(x+i \epsilon)\right). $$ It seems obvious that the support of $\rho$ is located on the cuts of $G$. What is not so obvious to me is how to see that this representation "works", i.e. that the RHS of the first equation is indeed equal to the LHS for all $x$ in the Riemann surface.

Starting from the right-hand side, I can see the following: $$ \int dy \frac{\rho(y)}{x-y} = \sum_k\int_{C_k} dy \frac{\rho(y)}{x-y} = \sum_k\oint_{C_k} dy \frac{G(y)}{x-y}, $$ where the $C_k$ are the cuts of $G$ and the last expression represents a sum of integrals along contours around the cuts $C_k$.

How do I proceed from here to prove that this is indeed a representation of $G(x)$?

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After some extra reading, I think I can answer this now, although not from the perspective I originally intended. We use the Plemelj-Sokhotksi formula, in the following way: in the limit $\epsilon \downarrow0$ we have $$ G(x-i\epsilon) -G(x+i\epsilon) = \int \left( \frac{\rho(y)}{x-y-i\epsilon} -\frac{\rho(y)}{x-y+i\epsilon}\right)dy = -\int \left( \frac{\rho(y+x)}{y+i\epsilon}-\frac{\rho(y+x)}{y-i\epsilon}\right) dy = i\pi \rho(x) +P\int \frac{\rho(y+x)}{y} dy +i \pi \rho(x) -P\int \frac{\rho(y+x)}{y} dy = 2\pi i \rho(x), $$ for all $x\in C$. The $P\int$-sign used here is the principal value integral in the Cauchy-sense. This derivation shows the consistency of the definitions of $G$ and $\rho$, which was what I was after.

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