We know that the integral
$$I = \int_{-\infty}^{\infty} \mathrm{d}x e^{-x^2}$$
can be calculated by first squaring it and then treat it as a $2-$dimensional integral in the plane and integrate it in polar coordinates.
Are there any other ways to calculate it? I know that we may use the relation
$$\Gamma(x)\Gamma(1-x) = \frac{\pi}{\sin{\pi x}},$$
but this, in effect, is still taking the square.
Well, after I write down the above text, I figure that maybe there is no way to calculate it without squaring, since, after all, the result contains a square root, and it seems no elementary function can "naturally" produce a square root of $\pi$ starting from natural numbers (though I don't know how to describe this more concretely; you are also welcome to comment on this point). Nevertheless I still post this question in case there are some other ideas.
EDIT: the Fourier transformation method at Computing the Gaussian integral with Fourier methods? appears kind of cheat to me, since the very proof of the Fourier transformation formula actually makes use of the value of the Gauss integral (at least in this wiki page http://en.wikipedia.org/wiki/Fourier_inversion_theorem#Proof).
Thank you.
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