منابع مشابه
The Maclaurin Expansions
The papers [15], [16], [4], [12], [2], [14], [5], [1], [3], [7], [6], [10], [11], [8], [9], [17], and [13] provide the notation and terminology for this paper. The following proposition is true (1) For every real number x and for every natural number n holds |x| = |x|. Let f be a partial function from R to R, let Z be a subset of R, and let a be a real number. The functor Maclaurin(f, Z, a) yie...
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a Bk({1− t}) k! f (t)dt where a and b are arbitrary real numbers with difference b − a being a positive integer number, Bn and bn are Bernoulli polynomials and numbers, respectively, and k is any positive integer. The condition we impose on the real function f is that it should have continuous k-th derivative. The symbol {x} for a real number x denotes the fractional part of x. Proof of this th...
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متن کاملThe Euler - Maclaurin Expansion
We have seen that the accuracy of methods for computing integrals or derivatives of a function f(x) depends on the spacing between points at which f is evaluated, and that the approximation tends to the exact value as this spacing tends to 0. Suppose that a uniform spacing h is used. We denote by F (h) the approximation computed using the spacing h, from which it follows that the exact value is...
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ژورنال
عنوان ژورنال: Edinburgh Mathematical Notes
سال: 1949
ISSN: 0950-1843
DOI: 10.1017/s0950184300002822