欧拉-麦克劳林公式(Euler-Maclaurin formula)是微积分中的一个重要公式,用于估计求和、积分或级数的近似值。该公式以两位数学家欧拉和麦克劳林的名字命名。 具体地说,给定一个函数f(x),欧拉-麦克劳林公式给出了如下的近似公式: $$\sum_{k=a}^{b} f(k) \approx \int_{a}^{b} f(x)dx + \frac{f(a)...
5) Euler-Maclaurin formula Euler-Maclaurin定理 1. This essay deduce s the power sum formula of an arbitrary arithmetic progression step by step on t he basis ofEuler-Maclaurin formula,and then achieres the compute formula of nat ural number power sum-∑mi=1im. ...
Euler-Maclaurin summation formulaasymptotic formulaeLet $n$ be a fixed positive integer. In this article we obtain asymptotic formulae for $sum^{N}_{j=1} j^n log j$ and $prod^{N}_{i=1}j^{(j^n)}$. For example if $n=1$ we obtain the asymptotic formulae,[sum^{N}_{i=1} ...
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The latter problem was connected to the question of expressing the sum of a series using an integral. The outcome of this research was Euler's derivation of what would later become known as the Euler-Maclaurin formula. Euler subsequently returned to interpolation and formulated the theory of ...
The Euler-Maclaurin formula for simple integral polytopes. We give a Euler-Maclaurin formula with remainder for the sum of a smooth function on the integral points in a simple integral lattice polytope. Our proof u... Y Karshon,S Sternberg,J Weitsman - 《Proceedings of the National Academy...
(3.6) We see below that the final exponential term is asymptotically negligible, and we next show that when this final term is removed, the resulting sum indeed gives the claimed asymptotic formula. Proposition 3.2 As t → 0+, we have m − 1 4 e2m 3 2 −m3 t ∼ et −1 m≥...
We give an Euler-Maclaurin formula with remainder for the weighted sum of the values of a smooth function on the integral points in a simple integral polytope. Our work generalizes the formula obtained by Karshon, Sternberg and Weitsman in the ''Euler-Maclaurin with remainder for a simple ...
[L])andtheBrianchon-Gramformula(see[Br],[G],[So]and[Gr])(cf.Re- mark38). Thewell-knownclassicalpolytopedecompositionformulaofBrianchon-Gramex- pressesthecharacteristicfunction1 P ofaconvexpolytopePasthealternatingsum ofthecharacteristicfunctionsofalltangentconestothefacesofP.Byflippingthe 2000...
Two sum formulas are given concerning these two numerical arrays. 给出了关于这两个数列的两个求和公式。 更多例句>> 5) summation formula 求和公式 1. The summation formula of series sum from k=2 to ∞ f(k)■(k); 级数sum from k=2 to ∞f(k)(非汉字符号)(k)的求和公式 2. Kratten...