I am dealing with a domain of integration of the form:
$\left(\frac{x-y}{x+y}\right)^2+\left(\frac{y-z}{y+z}\right)^2+\left(\frac{x-z}{x+z}\right)^2\leq k$
The region looks like this (for $k=0.2$):
It becomes just a line for the limit case $k=0$ and the full first octant when $k=3$.
All my efforts so far were unfruitful:
- Use Mathematica's "Reduce" command in order to get the limits (untreatable).
- Several different change of variables.
- Approximating it to an infinite pyramid/cone
- Profiting from the symmetry (also of my density function) and take only one third of this region, for instance intersecting it with $x - y \leq 0$ and $z - x \geq 0$. The idea was to be able to integrate there and just multiply the result by 3.
- Integrate only on the extended "shadows" (projections?) and meet the results afterwards.
- Truncate it with a plane $x+y+z<a$ with the intention to let $a\to\infty$ after integration
- And more efforts I'm ashamed to share.
Could you please provide me with some further ideas? Does anyone even know if this surface has a name? I would like to avoid going numerical. Is there a way to solve it numerically, but not completely? I need a sharp estimate of the final integral as a function of $k$.
The fact that is improper doesn't bother me, since the integrand is well defined for unbounded regions.
Update: I tried applying the divergence theorem, but the expression of the normal vector is incredibly discouraging (more than 125 lines in Mathematica). I doubt that I could integrate that (didn't even try).
Any ideas...?