Complex Analysis: Gaussian Integral

Today, we use a very exotic contour integration methods to evaluate the Gaussian integral.

[konchady]

What makes this proof elegant is also that you used the general rule of making contour integrals useful. That is, if you don't have poles or singularities, add the appropriate ones to get the result. This is itself an art.

[konchady]

Brilliant !! For most of my life, I was looking for a countour proof that didn't involve that annoying parallelogram contour. You cleverly changed it into something much more elegant and that is commendable.

[elahehghanati]

Thanks a lot. In my doctorate research I use Gaussian integrals as I work on laser beam propagation. Your solution now will be part of my thesis appendix and I will cite your name. I feel really better knowing this integral solution process. Do you know what will be the result for exp(-z*x^2), where z is complex number?

[digxx]

Did you come up with it yourself, or where is it from? Btw your argument about ib=tau being complex technically defines a parallelogram can be made a bit more rigorous for every finite R (large enough so that the pole is inside), by noting that you can deform the contour without crossing any singularity.

[Calcprof]

The value of τ makes the size of the "rectangle" slightly scew, so that the upper integral is not precisely from R to -R, but its only off a little, and in the limit it down't matter

[mustafaunal]

Thank you! This method is ingenious! Who discovered the idea?

[mihaigabrielbabutia]

When calculating the limit of the integral over gamma_2, do we have a guarantee that the limit and the integral can be interchanged for the argument? I was thinking of the dominated convergence theorem for Lebesgue integration here, that may be the key here.
Very interesting approach, I only saw the method of evaluation by double integral, but this is far more elaborate and pleasant to follow!

[arielfuxman]

30:10 Is it formal? Interchanging of an integral and a limit

[hectore.garcia]

What would justify your assumption that g(z)=g(z+tao), what leads you to think that this is actually the case?

[geremiasjunior]

Hello, how do you define the vertices of the rectangle, for example, I'm solving an integral whose denominator is cosh (pix / 2), how can I get the upper vertices, would it be something like R + ib, -R + ib?

[ldego_]

How would it be resolved if it were e^-(ax^2)?🦊

[dfk]

Hi! What about integrals INVOLVING Gaussian functions? I know by default it is not suitable for a contour enclosed at infinity, as the Gaussian diverges in both directions of the imaginary axis, but any tricks on that?

[bonelesspizza]

I think there might be a mistake at 19:18. how does dividing by sqrt(pi*i) on both sides give sqrt(pi*i)? you gave tao/2 but it would be pi*i/(2)(sqrt(pi*i) and then the odd integer. could I get why we got that fraction instead?

[ian]

When do I use a rectangle as a contour?

[wuyqrbt]

I want to learn complex analysis, you think this example (Gaussian integral) is good place to start or you have another suggestions for me?
(And I should tell you I'm familiar with this kind of thing (complex analysis), and I just to practice more and ... )
Thank you

[aweebthatlovesmath]

Hey can't you say integral over gamma 2 is equals to mines integral over gamma 4 so integral over gamma 4 is negative zero so zero.

[yateendrasihag]

Hey thanks for the video but can you tell me how do you write "exp(sqrt(i))=exp(i*pi/4)" at 23:13 ?

[dynastieeyala]

Si seulement. Il expliquait tout ça en français j'aillais peut-être comprendre