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So upon thinking about this more and playing with different assumptions when trying to integrate the above function(s), what I found was the following - $\int_{a}^{\infty} dt \frac{e^{i k t}}{t + i \tau} = e^{k\tau}\,\Gamma\left(k(-ia + \tau) \right)$, where $\Gamma$ is the incomplete Gamma function. Now, suppose I try doing this integral with ...