Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

i am currently reading through a book on generalized functions, and there it is said that:

... $\int_{|x|\le r} |x|^{-t} dx$ converges for $t < n$ (in $n$ dimensions) and diverges for $t \ge n$.

Why does it converge for $t < n$, its area/volume still goes to infinity near zero?

share|improve this question

1 Answer 1

up vote 2 down vote accepted

Use spherical coördinates. The $c_n$ shown below is the surface area measure of $S^{n-1}$.

$$\int_{B_r(0)} |x|^{-t}\, dx = \int_{S^{n-1}}\int_0^r \rho^{-t} \rho^{n-1}\,d\rho\,d\sigma = c_n \int_0^r \rho^{n-1-t}d\rho$$

The last integral converges iff $n-1-t > -1$, or if $n > t$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

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