If, let's say, I have an integral such as $$\int_{-\infty}^{\infty}\frac{dt}{(t-a)(t-b)(t-c)(t-d)}$$ where $a$, $b$, $c$ and $d$ are complex numbers such that the poles falls in all the four quadrant and $t$ is time.
Now to do the integral what contour should I choose? Also which poles should I keep inside the contour and which should I keep outside so that causality is maintained?
I know for the case when we have an exponential in the denominator we generally choose the contour considering the side for which the exponential decays to 0.
Though it is a very general problem but is there any way to tackle them? For this problem, it seems all the possible combination of poles stays inside or outside the contour would produce different results.
Some help in this regard will be highly appreciated.
I think you can choose a contour by closing from above or below (and show that in the limit of large radius, the contribution of the circle vanishes), excluding the poles at a, b, c and d. Use the Residue Theorem (just calculate simple derivatives, instead of integrating by parts). See https://en.wikipedia.org/wiki/Residue_theorem.