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seen Dec 4 at 5:52

Jan
14
accepted Asymptotic behavior of an integral
Jan
13
comment Asymptotic behavior of an integral
Sure, but it makes the $\epsilon$ divergence worse, because this comes from small $x$.
Jan
13
answered Asymptotic behavior of an integral
Jan
13
comment Asymptotic behavior of an integral
@Zarrax, I tried this, but doesn't differentiating the denominator actually make the divergence worse instead of better? Am I missing something?
Jan
13
comment Asymptotic behavior of an integral
As a simple counterexample, take the integral $\int dx_1\, dx_2\, e^{-x_1-x_2}/(x_1+x_2)^2$, where the range is from $\epsilon$ to $\infty$. We can make the orthogonal change of variables $y_1=x_1+x_2$ and $y_2=x_1-x_2$, and the integral now becomes $\int dy_1\, dy_2\, e^{-y_1}/y_1^2$, which would seem to diverge by your argument. This integral is explicitly convergent though, so $y_2$ must enter through the bounds.
Jan
13
comment Asymptotic behavior of an integral
Hmmm I don't think so... $y_4$ still enters through the bounds on the other integrals, so I don't think it diverges.
Jan
12
comment Asymptotic behavior of an integral
This method looks like it will work. I'll give it a shot and accept this answer if it does.
Jan
12
comment Asymptotic behavior of an integral
@IgorRivin, in your improved answer I still get a Jacobian with zero determinant. However $x_4=y_4$ seems to work.
Jan
12
awarded  Critic
Jan
12
awarded  Supporter
Jan
12
comment Asymptotic behavior of an integral
I haven't tried that yet, I'll look into that. Thanks.
Jan
12
comment Asymptotic behavior of an integral
I tried that, but unfortunately the Jacobian determinant is zero for that transformation.
Jan
12
comment Asymptotic behavior of an integral
Thanks for the tip, I simplified this.
Jan
12
asked Asymptotic behavior of an integral
Oct
3
awarded  Scholar
Oct
3
accepted Lax Pairs for Linear PDEs
Jan
9
awarded  Student
Jan
9
asked Lax Pairs for Linear PDEs