Finding all Dimensions where $\frac{1}{|x|}$ is locally integrable

Question

I am trying to find all the dimensions in which $\frac{1}{|x|}$ is locally integrable, i.e.: $$\int...\int_{B(\textbf 0,\textbf 1)}u(\textbf x)d\textbf x<\infty$$ Where number of integrals denotes which space we are in. Now, for Dimension $1$, I got: $$\int_{-1}^0-\frac{1}{x} dx+\int_0^1 \frac{1}{x} dx$$ Which is not bounded, thus not locally integrable. Now I computed it in Dimension $2$: $$\iint_{D(0,1)}\frac{1}{|x|} dx = \int_0^1\int_0^{2\pi} \frac{1}{r} r drd\theta$$ The answer is $2\pi$. Now for Dimension $3$, I got $4\pi$ by Spherical Coordinates. But I used Spherical and Polar Coordinates for these. I have no idea how to generalize this for $n$ dimensions, unless they go up by a multiple of $2$ each time, which I sincerely doubt. Can someone help?

Answer 1

We can use hyperspherical coordinates in any number of dimensions. The integration measure can be expressed as $r^{d-1}drd^{d-1}\Omega$ where $d^{d-1}\Omega$ is the volume form for the unit sphere $S^{d-1}.$ So your integral will be $$ \int_0^1 r^{d-2}\int d^{d-1}\Omega = \frac{1}{d-1}V_{d-1}$$ where $V_{d-1}$ is the volume of the unit $(d-1)$-sphere (i.e. the surface area of the $d$-dimensional unit ball). We have $$ V_{d-1} = \frac{2\pi^{d/2}}{\Gamma(d/2)}$$ where $\Gamma$ is the Euler Gamma function. The proof of this formula through calculating the Gaussian integral in two different ways is neat and can be found in many places.