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?

  • $\begingroup$ The Feynman $i\delta$-regularization in eq. (2.39) is essentially what is done in part II of my Phys.SE answer here. $\endgroup$ – Qmechanic Sep 9 '19 at 16:41
  • $\begingroup$ I saw it. I hope someone can give a clearer answer along the same lines. $\endgroup$ – user2820579 Sep 9 '19 at 17:04

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