Timeline for How to analytically evaluate this n-dimensional iterated integral?
Current License: CC BY-SA 3.0
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| Jan 25, 2016 at 23:35 | comment | added | Iosif Pinelis | Using the paper by Plackett at jstor.org/stable/2332716?seq=1#page_scan_tab_contents (which, in particular, contains a reference to Schlafli) and a regularization of the kind you mentioned, one may be able to obtain a recursion reducing your $n$-fold integral to ones over sectors of smaller dimensions. | |
| Jan 25, 2016 at 22:58 | comment | added | Alexandre Eremenko | @Andrea Becker: that was exactly what I asked: in what sense do you want to understand your integral? Only absolutely convergent integrals are defined unambiguously. This one is not absolutely convergent. | |
| Jan 25, 2016 at 22:58 | history | edited | Andrea Becker |
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| Jan 25, 2016 at 22:55 | comment | added | Marcel | Its not divergent. If $a$'s and $b$'s are supposed to be large, you could think about a stationary phase approximation. I doubt it has an analytic evaluation in general. What I meant previously is that being non-obvious doesn't make it research level, I think. But I could be wrong, I guess. | |
| Jan 25, 2016 at 22:42 | comment | added | Andrea Becker | @Alexandre Eremenko For n = 1 and n = 2, I could evaluate it and the answer is finite, i.e. the integral is not divergent! If it helps you, think of it in a distributional sense, and regularize it by adding in the exponential $-\eta_k x_k^2$ and in the end take $\eta_k$ to $0+$. | |
| Jan 25, 2016 at 22:35 | comment | added | Alexandre Eremenko | But it is divergent. What do you mean by "evaluate"? | |
| Jan 25, 2016 at 22:32 | history | edited | YCor |
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| Jan 25, 2016 at 22:27 | comment | added | Yemon Choi | @Marcel Were you thinking of the definite integral (which would be easy)? This iterated one does not look obvious to me even for $n=2$, but perhaps I am missing something | |
| Jan 25, 2016 at 22:24 | history | edited | Andrea Becker | CC BY-SA 3.0 |
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| Jan 25, 2016 at 22:14 | comment | added | Andrea Becker | @Marcel If it's not of research level, what is the answer?! | |
| Jan 25, 2016 at 22:09 | comment | added | Marcel | your question is not of research level. You should ask in Math Stack Exchange | |
| Jan 25, 2016 at 22:07 | history | asked | Andrea Becker | CC BY-SA 3.0 |