I am trying to give a meaning to this integral
$$I(t)=\lim_{\epsilon\rightarrow 0^+} \int_0^t\frac{\phi(y)}{(\epsilon +iy)^n}\,dy$$
where, $n > 1$, $\phi(y)$ is a complex function, infinitely differentiable, and $y\in \mathbb R$. I thought it might be related to derivatives of $\delta(y)$ distributions, but I am quite stuck with the calculation.