Recently encountered an integral of the following kind: $$ \int\limits_{-\infty}^{\infty} \dfrac{1}{t - i \alpha} \dfrac{1}{t + i \beta/2} \dfrac{1}{(\tau - t) - i\alpha} dt ,$$ where $\alpha, \beta, \tau$ are real, $\alpha$ can be either positive or negative in general, but $\beta$ is always positive. In a sense, it is a convolution of a product of the first two terms with the third one. Expanding in a $\dfrac{C_1}{t - i \alpha} + \dfrac{C_2}{t + i \beta/2} + \dfrac{C_3}{(\tau - t) - i\alpha}$ does not help as the system on the constants yields the same product of $3$ fractions.
Any chances to have an analytical answer?
Side remark: With complex analysis tricks, I am a little bit puzzled... Even though for a given sign of $\alpha$ either the 1st or the 3rd term has a pole in an upper half-plane, and asymptotically the integrand behaves as $~1/k^3$ (so, seems integrable), the numerical answer does not match the analytical one, probably, I am missing something. Is that possible to use complex analysis tricks at all for such integrands?