Euler - Maclaurin Formula
نویسندگان
چکیده
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 theorem using h−calculus is given in the book [Ka] by Victor Kač. In this paper we would like to discuss several applications of this formula. This formula was discovered independently and almost simultaneously by Euler and Maclaurin in the first half of the XV III-th century. However, neither of them obtained the remainder term
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