I would very much appreciate any suggestions and/or pointers to references relevant for the analytic evaluation of the following n-dimensional iterated integral $$\int_{-\infty}^{+\infty}dx_1\int_{-\infty}^{x_1}dx_2\ldots\int_{-\infty}^{x_{n-1}}dx_n f_1(x_1)f_2(x_2)\ldots f_n(x_n)$$ where $$f_k(x_k) = \exp [i a_k x_k^2 +i b_k x_k]$$ and $a_k$, $b_k$ are arbitrary non-zero real numbers, and $i^2 = -1$.
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1$\begingroup$ your question is not of research level. You should ask in Math Stack Exchange $\endgroup$– MarcelCommented Jan 25, 2016 at 22:09
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2$\begingroup$ But it is divergent. What do you mean by "evaluate"? $\endgroup$– Alexandre EremenkoCommented Jan 25, 2016 at 22:35
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1$\begingroup$ @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 $0+$. $\endgroup$– Andrea BeckerCommented Jan 25, 2016 at 22:42
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1$\begingroup$ @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. $\endgroup$– Alexandre EremenkoCommented Jan 25, 2016 at 22:58
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1$\begingroup$ 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. $\endgroup$– Iosif PinelisCommented Jan 25, 2016 at 23:35
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