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I have a question about this integral:

$$ \int_{-\infty}^\infty e^{ikx^2} \, dx = \sqrt{\frac{\pi}{8}}(1+i) $$

Essentially we are following this curve with -- the Cornu spiral:

  • $x = \cos t^2$
  • $y = \sin t^2$

The Wikipedia article has an image, but I have some doubts.

  • Does the red spiral really converge to the blue point?
  • Or does it just approach a limiting circle with the blue point at the center?


See also: Orange Peels and Fresnel Integrals arXiv:1202.3033

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  • $\begingroup$ Your integral equals, when $\Im\left[\text{k}\right]>0$: $$\mathcal{I}\left(\text{k}\right)=\int_{-\infty}^\infty e^{i\text{k}x^2}\space\text{d}x=\frac{\sqrt{\pi}}{\sqrt{-\text{k}i}}$$ $\endgroup$ Sep 27, 2016 at 13:01
  • $\begingroup$ The parametrisation of the curve is $$t \mapsto \int_0^t e^{ix^2}\,dx.$$ Since the limit as $t \to \pm\infty$ exists … $\endgroup$ Sep 27, 2016 at 13:30

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On the wikipedia page it is proved that $$\int^{\infty}_0\sin x^2 = \int^{\infty}_0\cos x^2 = \sqrt \frac \pi 8$$ so yes.

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I have just (24-Mar-17) posted an answer to another Cornu spiral question here: Is this Cornu spiral positively oriented or not?. It provides a closed-form analytic solution for the Cornu spiral and it is evident that the spiral does indeed approach the blue dots, albeit very slowly.

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