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)...
和"臭名昭著”的Euler-Maclaurin summation formula(欧拉-麦克劳林求和公式) mathologer: 如何计算幂和1^k+2^k+...+n^k, 伯努利数与欧拉-麦克劳林公式,巴塞尔问题_哔哩哔哩_bilibili 话说这个欧拉-麦克劳林求和公式和巴塞尔问题有什么关系吗?Searching relevant knowledge in papers: Euler-Maclaurin summation formula...
Euler-Maclaurin formulaPoisson's formulaIn q -calculus, the Jackson formula (19.2) provides a way to compute explicitly a q -antiderivative of any function. Recall that the Jackson formula was deduced formally using operators. We will do a similar thing for the h -antiderivative in this ...
摘要: In q -calculus, the Jackson formula (19.2) provides a way to compute explicitly a q -antiderivative of any function. Recall that the Jackson formula was deduced formally using operators. We will do a similar thing for the h -antiderivative in this chapter....
THE WEIGHTED EULER-MACLAURIN FORMULA FOR A SIMPLE INTEGRAL POLYTOPEJosé AgapitoJonathan WeitsmanWe give an Euler Maclaurin formula with remainder for the sum of the values of a smooth function on the integral points in a simple integral polytope. This formula is proved by elementary methods....
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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[L])andtheBrianchon-Gramformula(see[Br],[G],[So]and[Gr])(cf.Re- mark38). Thewell-knownclassicalpolytopedecompositionformulaofBrianchon-Gramex- pressesthecharacteristicfunction1 P ofaconvexpolytopePasthealternatingsum ofthecharacteristicfunctionsofalltangentconestothefacesofP.Byflippingthe 2000...
2006年10月 COLLEGEMATHEMATICS Oct.2006 关于Euler2Maclaurin公式的一个注记 钟尔杰, 黄廷祝 (电子科技大学应用数学学院,成都610054) [摘要]利用梯形公式的余项,将被积函数的二阶导数做幂级数展开,证明了余项是关于求积区间长度 的奇数次幂级数.推导出了复合梯形公式的一类渐近展开式,从另一方面印证了Euler2Maclaurin公...
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 inexplicable functions including the gamma function. The methods used by Euler illustrate well the principles ...