i had a moment of nostalgia flipping through this. i remember being shown how to do this integral (using the polar coordinates method) freshman year in undergrad by a theoretical physicist that dramatically overestimated my abilities lol ("just complete the square duh").

sorry don't have much substantive to say, other than that it's interesting you can have a feeling of nostalgia from something as abstract as this.

I think the tenth (Cauchy residue) and eleventh (Fourier domain) are my favourites, and what I'd use if I had to produce a proof to save my life.

I didn't know you could use the Stirling Approximation though. Anyone who does any thermodynamics should be very familiar with this trick, but I think people rarely include the second order term with the suspicious sqrt(pi) factor, much less think about where it comes from.

You could probably perform some sort of math personality test by asking which proof the respondent likes best! I was particularly charmed by #5 (construct a volume of revolution, then slice it two ways).

For me it's #4. I think this was a trick Feynman used a lot. I always like seeing intimidating problems solved simply with an uncommon tool.

I also liked 4. I remember this being a trick question on a calc exam right after we learned differentiation under the integral.

A nice anecdote about Poisson's proof (the double integral + polar coords trick): Grothendieck, widely considered the greatest mathematician of the 20th century didn't know about the proof [1]

Might an urban legend, but it's still interesting.

[1] https://hsm.stackexchange.com/questions/5977/grothendieck-an...

I think this is entirely plausible. Grothendieck did not walk the established paths, but invented lot's of new concepts as he was working on problems. IIRC he was largely an auto-didact in his early years. In his work he never tried to reconcile the existing theory, and somehow improve it. He started form a blank slate (see EGA/SGA/FGA series of books) and build-up the proper concepts (Schemes,K-theory,Categories) that would allow the problem to unfold in the most natural setting and slowly resolve itself: https://ncatlab.org/nlab/show/The+Rising+Sea

Sometimes ignorance is a blessing. It frees you from keeping all the historic clutter in your memory, and see an un-distorted picture.

Grothendieck worked in algebra, this is analysis.

He started his career as a functional analyst, actually, but yes he was an algebraist by inclination.

If people hadn't invented a method to do Gaussian integrals, entire subjects like statistical mechanics and quantum field theory would have been nonexistent.

in this video another way to calculate the integral

https://www.youtube.com/watch?v=X_1uiQhGOPU