Giovanni Ferraro, Some aspects of Euler's theory of series: Inexplicable functions and the Euler-Maclaurin summation formula, Hist. Math. 25 No. 3, 1998, 290-317.Ferraro, G.: Some aspects of Euler's theory of series: Inexplicable functions and the Euler-MacLaurin summation formula. Historia ...
The Euler2Maclaurin summation formula , the sampling theorem , and approximate integration over real axis. Linear Algebra and its Applications , 1983 , 52253 (1) :144Butzer, P.L., Stens, R.L.: The Euler–MacLaurin summation formula, the sampling theorem, and approximate integration over the ...
When the Clenshaw-Curtis formula is used in a way different from that put forward by the original authors it is suggested that it may have significant advantages over other methods of numerical integration in many problems. 展开 DOI: 10.1007/BF01436254 被引量: 9 年份: 1965 ...
Ingham also discussed a special case that eliminates the need for calculating an exact asymptotic formula for f throughout the complex domain D (although it is still necessary to bound the asymptotic order of f as in condition (ii)). The following result is [13, Theorem 1 ]. Theorem 4.2 ...
The Euler-MacLaurin Sum FormulaLet f ( x ) be continuous with as many continuous derivatives as required. Noticing B ′ 1 ( x ) =l we obtain through integration by parts $$\\int\\limits_0^1 {f(x)\\,dx\\, = \\,[{B_1}(x)f(x)]_0^1 - } \\int\\limits_0^1 {{B_1}...
integrationIn this short note we prove an extension of the Euler–Maclaurin expansion for general rectangular composite quadrature rules in one dimension when the derivative of the integrand has a logarithmic singularity. We show that a correction series has to be added to the formula, but that ...
product integrationFermi-DiracintegralsThe present paper deals with a generalization of the Euler-Maclaurin summation formula. The generalization is based on Bernoulli functions which are expressed in an integral form involving Bernoulli polynomials. Then the formula is used to numerical computation of the...
The problem of interpolation led Euler to formulate the problem of integration, i.e., to express the general term of a series by means of an integral. The latter problem was connected to the question of expressing the sum of a series using an integral. The outcome of this research was ...
integration/ Hadamard finite-part integralsEuler-Maclaurin formulaasymptotic behaviourweakly singular integralsparametric sigmoidal transformationCauchy principal valueerror analysis/ A0260 Numerical approximation and analysis A0230 Function theory, analysis
integration/ generalized Euler-Maclaurin formulatrianglesasymptotic expansioncomposite integration formulaebivariate functionaffine mapsSobolev spaces/ C4260 Computational geometry C1160 Combinatorial mathematics C4160 Numerical integration and differentiationWe prove the existence of an asymptotic expansion of the ...