I would like to perform the following integral
\begin{equation} \int_{-\infty}^\infty dz \ \frac{e^{-i zt} \sqrt{-i z}}{\sqrt{-i z} + c} \end{equation}
which corresponds to the inverse Fourier transform of some response function I am interested in. I have tried with a change of variable $-iz \to -\eta^2$, which should simplify the denominator so that I only have to deal with a simple pole. However, I am having issues with deciding what contour to integrate over (the integral is no longer to be carried out along the real axis). This is because of the new exponential term $e^{-\eta^2 t}$, which only vanishes for positive $t$ and $|\eta|^2\to \infty$ if $\Re(\eta) > \Im(\eta)$. This seems to prevent me from being able to enclose the pole in the contour. I don't expect this integral to be zero so I am probably doing something wrong here. Any idea?