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Why does the following integral converge?

\begin{equation} \int_{0\leq |x_1|\leq|x_2|\leq\dots\leq |x_n|\leq 1}\frac{(1-\sum_{i=1}^n\log |x_i|)^r}{|x_1x_2\dots x_n|^{1/2}}dx_1dx_2\dots dx_n \end{equation}

Here, $r$ is any positive real number.

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1 Answer 1

up vote 4 down vote accepted

You can show (for example, using l'Hôpital's rule) that $\frac{(1-\log(t))^r}{t^{1/2}}$ is dominated by $\frac{1}{t^{2/3}}$ for sufficiently small $t$. If you use this along with the fact that that $\sum_{i=1}^n\log|x_i|=\log|x_1\cdots x_n|$, you have your integrand dominated by $|x_1\cdots x_n|^{-2/3}$ in a neighborhood of the set where it is undefined, and bounded on the complement of this neighborhood. Does that get you where you want to be?

Also, to keep things simple, you may as well just consider $0\leq x_1\leq x_2\cdots\leq x_n\leq1$, because the whole integral can be broken up into $2^n$ equal pieces depending on the various choices of sign.

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Indeed. Thanks very much! –  iamtheneal Oct 6 '10 at 21:49
    
You're welcome. –  Jonas Meyer Oct 6 '10 at 21:57

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