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I have a function defined as follows:

$$M(x) = \int_{-\pi}^{\pi} dq \frac{1}{(\Omega - f(x_0 - q)) - x}$$

Here, $f(x)$ is a non-negative analytic function, with $f(0) = 0$. For $x \ge \Omega$, it is possible to have a singularity in the integrand.

(The motivation: The expression arises in Wigner-Brillouin perturbation theory to compute corrections to the energy of a quantum state in the Holstein model)

What I need is the asymptotic behavior of $M(x)$ as $x$ approaches (and exceeds) $\Omega$, but I am unsure about how to start here. The singularity occurs at $x_0 = q$ for $x \rightarrow \Omega$, and I can certainly expand $f$ around that point, which yields something like $A\cdot x^2$. But I am not sure how to use that inside of the integral.

For my special case I am assuming a simple dispersion relation, $$f(x) = 2t \cdot (1 - \cos(x))$$. According to the physical literature, the asymptotic behavior should be of the type $c \cdot (\Omega - x)^{-1/2}$.

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