How is this step justified into calculating Gaussian integral?

Question

Can anyone give me the justification for the following step in the Gaussian integral evaluation.

$$I^2= \int_{\infty}^{\infty}e^{-x^2}dx \int_{\infty}^{\infty}e^{-y^2}dy=\int_{\infty}^{\infty} \int_{\infty}^{\infty}e^{-(x^2+y^2)}dxdy$$ Is there some sort of proof I can look at to see why it should be the case that you can combine the two integrals into a double integral.

Note that there are numerous answers to the question "Proving $\int_{0}^{\infty} \mathrm{e}^{-x^2} dx = \dfrac{\sqrt \pi}{2}$", although not necessarily one that explains this particular step of the proof in the detail you want. — David K, Jun 04 '15 at 20:53

score 0 · Accepted Answer · answered Jun 04 '15 at 20:37

0

It is enough that these integrals are independent of each other (one is in $x$, the other in $y$). You can see $\int e^{-x^2} dx$ as a constant with respect to integrating in $y$.

answered Jun 04 '15 at 20:37

Tom Ultramelonman

642

Okay thanks. The other comment was helpful also. They are both similar responses. – Sam Jun 04 '15 at 20:42

How is this step justified into calculating Gaussian integral?

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