1
$\begingroup$

Even if I've red other threads treating this question, it's still obscure to me what deeply relates the multiple Guassian integral $\int e^{-x^2} = \sqrt \pi$ and the area of a $n$-ball.

Someone could offer an insight?

$\endgroup$
2
$\begingroup$

$$ \pi^{n/2} = \int \exp \left( -\sum_{i=1}^n x_i^2 \right) \mathrm d V = \int_0^\infty \exp \left( -r^2 \right) r^{n-1} \mathrm d r \int \mathrm d \Omega. $$ The second equality is change of cartesian coordinates into spherical and application of Fubini theorem. $\int \mathrm d \Omega$, which is also the surface of a sphere in $n$ dimensions of unitary radius, can be computed from this relation. This is because $\int_0^\infty \exp \left( -r^2 \right) r^{n-1} \mathrm d r = \frac 12\Gamma \left(\frac n 2 \right)$ and we know now to deal with Gamma function.

Knowing how to compute $\int\mathrm d \Omega$ you can compute the volume easily. Once again using spherical coordinates and Fubini theorem $$ \int_{r<R} \mathrm d V = \int_0^R r^{n-1} \mathrm d r \int \mathrm d \Omega. $$

$\endgroup$
  • $\begingroup$ Thank you. Just a question: $\int \mathrm d \Omega$ gives the surface area of a $(n-1)-$sphere of unitary $r$? $\endgroup$ – Lo Scrondo Mar 5 '16 at 13:40
  • $\begingroup$ @LoScrondo Yes, you are right. I'll edit the answer to fix this. $\endgroup$ – Korf Mar 5 '16 at 13:43

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.