Suppose $a(z) = \sum_{k=0}^n c_ke^{i\alpha_k z}, z\in\mathbb{C}$ for some real numbers $\alpha_k$. What methods would be available to me if I wanted to look at the asymptotic behaviour of $$I(s) = \int_a^b \dfrac{e^{i \xi t}}{|a(t)|^2}\,dt$$ as $\xi \to \pm\infty$? I did some computations (similar to this computation on asymptotics of Fourier coefficients) using residues for the simple case where $n = 1$ and $[a,b]$ matching the period of $|a(t)|^2$ but is it doable for $n>1?$
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