[drbaskin]

I'm not a big fan of introducing a new constant (though I believe \pi should have been 2\pi), but I love thinking of the integral you wrote down as \sqrt{\tau / 2} because then the answer practically tells you how to derive it!

How to derive the value of the integral: Square the integral to make it an integral in two variables, introduce polar coordinates, then change variables.

[Jach]

Yes, it's one of my favorite proofs. (I think I like it more than Euler's formula, especially since many calc teachers will look at e^{-x^2} and say it's un-indefinite-integrable without a second thought at what else it can do.)

But I'm not quite sure how you seeing it as \sqrt{\tau/2} helps you see the proof more easily. Because if you see \tau (ignoring the 1/2) you think "It has to do with circles or polar form." as per my rule of thumb?

[drbaskin]

I'm sorry I wasn't more clear, but your interpretation is what I meant. Per your rule of thumb, seeing \tau should suggest that it has to do with circles or polar coordinates, and the square root points to how to get the polar coordinates.