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I'm unable to find an easy way to compute the following multiple definite integral. Apologies if it is trivial.

Let $C$ be a $N \times 1$ real vector. Let $M$ be a $N \times n$ real matrix. Let $\left\| {} \right\|$ be the Euclidean 2-norm. Let $m \in \mathbb{N}$ sufficiently large for the integral to be convergent.

Compute $\int\limits_0^{ + \infty } {{{\left\| {C - MX} \right\|}^{ - m}}{d^n}X} $

Thank you for your help.

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What makes you think that this integral could have a "nice" answer? What would you want to do with the answer if you had it? –  Igor Khavkine Oct 24 '12 at 12:19
I assume the notation $\int_0^{+\infty}\ldots d^n X$ just means you're integrating over all space. What attempt have you made at a solution? Have you considered, for example, the case where $N=n$ and $M$ is diagonalizable? Have you tried a series expansion of the integrand when $m$ is even? –  Ben Crowell Oct 24 '12 at 12:36
@Ben. Right: this is a $n$-dimensional integral. Unfortunately, I'm not considering the special case $N = n$. If $n$ is even, then this is a multivariate rational fraction integral, the reason why I hope there is a closed-formed formula but I am not sure. If m is odd, we may need to make hyperbolic changes of variables. –  Pascal Orosco Oct 24 '12 at 12:57
@Pascal, the notion of "nice" depends entirely on your expectation when posing this question, that's why it would be helpful to have that clarified. Otherwise, you could consider just the expression for this integral to be its own evaluation, but I'm sure that's not very helpful to you. Also, for some fixed choice of $C$, $M$, $N$, and $n$, the integral could be straightforwardly approximated numerically (for instance, using Monte Carlo integration). However, that may not be useful to you either, if you expect to have some analytic information about the dependence of the answer on parameters. –  Igor Khavkine Oct 24 '12 at 14:43
Integrating over the entire space is trivial if you still remember some linear algebra and the change of variable formula and do not mind precomputing a few fixed integrals. Unfortunately, I suspect that it is the positive octant, which is the integration domain. That is much harder... –  fedja Oct 24 '12 at 19:20

1 Answer 1

This is certainly not a complete answer, but only some suggestions on how to reduce the complexity of the problem.

It does not seem to me like the possible rationality of the integrand, depending on the value of $m$, would be of great help. Integrals of rational functions are likely to produce non-rational ones (logarithms, roots, etc.). So computing the coordinate integrals one by one seems hopeless. Also, using global methods, like residues also seems problematic because the integration domain is restricted to the first orthant.

On the other hand using the SVD decomposition of $M$ puts the integrand into canonical form, where the square of the norm becomes a sum of squares, without cross terms between the integration coordinates. The downside is that the integration domain is no longer the first orthant, but rather a rotated and stretched version thereof.

The level sets of this new integrand are then concentric spheres, with center determined by the vector $C$. This symmetry can be exploited by changing to spherical coordinates with the same center. The Integrand then becomes $\int_0^\infty \mathrm{d}r~ r^{n-1} (r^2+c^2)^{-m/2} \int_{P_r} \mathrm{d}\Omega$, where $c^2$ is the least squares residual determined by $C$, $\mathrm{d}\Omega$ is the angular integration $P_r$ is the curved $n$-simplex obtained by intersecting the $n$ planes delimiting your integration domain with the sphere of radius $r$. The entire complexity of the integration reduces to computing the volume of $P_r$ as a function of $r$.

Unfortunately, this volume computation is in itself likely to be rather non-trivial. I do know that the computation of volumes of spherical simplices is a classical subject, treated early on by Schläfli. However, not being very familiar with the subject, I wouldn't know how to find the exact formulas. This survey article by Vinberg may be of help, though. An extra complication is that your $P_r$ are not precisely spherical simplices, because their sides do not correspond to higher dimensional "great circles", unless one of the planes delimiting your integration domain passes through the origin. However, the $P_r$ do approach true spherical simplices in the limit of large $r$. If you were able to find a decent analytic approximation for the volume of $P_r$ for small $r$, you may be able to get approximate analytic control over the whole integral. But I don't know how to do that.

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@Igor. Thank you very much for this approach to my problem. Precisely, I have not yet been able to find any nice reference in the literature about this kind of integral. So reducing the problem to another one is a good starting point. I'll try to follow your way... –  Pascal Orosco Oct 25 '12 at 14:33
@Igor. But as you said, this is not a genuine answer to my question, just one possible way to find such an answer. We still do not know if we can get closed-formed formulae in terms of special functions for instance. –  Pascal Orosco Oct 26 '12 at 7:41

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