# Gaussian oscillatory integral evaluation using regularization

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?

• The Feynman $i\delta$-regularization in eq. (2.39) is essentially what is done in part II of my Phys.SE answer here. – Qmechanic Sep 9 at 16:41
• I saw it. I hope someone can give a clearer answer along the same lines. – user2820579 Sep 9 at 17:04