To evaluate the Gaussian integral
$$ \int_{-\infty}^\infty dx e^{iax^2} = \sqrt{\frac{\pi i}{a}}, $$
one can use an appropriate contour as here, or use the method of "regularization", contained for example in Ashok p24, Eq. (2.39):
$$ \int_{-\infty}^\infty dx e^{\frac{im}{2\hbar \epsilon}x^2} = \operatorname{lim_{\delta\rightarrow 0^+}} \int_{-\infty}^\infty dx e^{(\frac{im}{2\hbar \epsilon}-\delta)x^2} = \sqrt{\frac{2\pi i\hbar\epsilon}{m}}.\tag{2.39} $$
Once one introduces this new parameter $\delta$, how the integration is carried out? Is it that I have to perform the integral over the complex plane as in the first case?
