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I want to approximate a multiple sum of the form $$\sum_{x_1+x_2+\cdots+x_m \leq n}e^{g(x_1,x_2,\ldots,x_m)},$$ where each $x_i$ is an integer between $0$ and $n$, by an integral $$\int_{x_1+x_2+\cdots+x_m \leq n}e^{g(x_1,x_2,\ldots,x_m)}dx_1dx_2\cdots dx_m\,.$$ I know that the Euler-Maclaurin formula can be used to derive the error term when $m=1$ but often see sums of this form with $m > 1$ approximated by integrals, though with little justification. I do not have much of a background in mathematical analysis so am not sure where to look for a reference for this.

Any help will be much appreciated.

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In fact Euler-Maclaurin applies to your case as well. See – Steve Huntsman Apr 2 '10 at 17:51
First of all, for a "generic" function $g$ your approximation by sum can be really bad. There are different techniques in asymptotic analysis (including Euler-Maclaurin) which can be used for your sums. – Wadim Zudilin Jun 9 '10 at 11:49

See Chapter 10 of Computing the Continuous Discretely by Beck and Robins for an introduction, and the paper of Brion and Vergne cited in the link in my comment for the general case.

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Thanks Steve. Looks like a very interesting book. BTW, the google books view is limited but there's actually a full, non-printable, .pdf on the author's website – bandini Apr 3 '10 at 10:54

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