A Further Extension of the Euler-Maclaurin Summation Formula
A Further Extension of the Euler-Maclaurin Summation Formula
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DOI：
10.1002/sapm1962411155
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发表时间：
1962-04
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影响因子：
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通讯作者：
I. Navot
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文献类型：
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作者：
I. Navot
在（1）和（3）中，n，m是任意正整数，f（x），g（x）是任意真实的连续函数，在0;;：; x;;：; 1处导数连续到2 m阶. a通常受0< a;;：; 1限制，但两个最重要的实际情况是a=！a= 1。BI ′（a）和r（s-j，a）分别是p次Bernoulli多项式和广义Riemann zeta函数。R2 m和p2 m是相关的导数，都是（1/n）2 m-I的阶，由下式给出：
In (1) and (3) n, m are arbitrary positive integers andf (x), g (x) are arbitrary real continuous functions, with continuous derivatives up to the order 2m, at 0;;:; x;;:; 1. a is generally restricted by 0< a;;:; 1 but the two most important practical cases are a=! and a= 1. BI'(a) and r (s-j, a) are Bernoulli's polynomial of degree p. and the generalized Riemann zeta function respectively. R2m and p2m are the pertinent remainders, both of the order of (1/n) 2m-I, given by