A generalization of the Euler-Maclaurin summation formula: an application to numerical computation of the Fermi-Dirac integrals

DOI10.1007/s10915-007-9175-3zbMath1203.65013OpenAlexW1968531005MaRDI QIDQ618347

Grzegorz Rządkowski, Sławomir Łepkowski

Publication date: 16 January 2011

Published in: Journal of Scientific Computing (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s10915-007-9175-3

zbMATH Keywords

numerical integrationproduct integrationBernoulli polynomialsBernoulli functionsfermi-Dirac integralsgeneralized Euler-Maclaurin summation formula

Mathematics Subject Classification ID

Computational methods for problems pertaining to quantum theory (81-08) Other functions defined by series and integrals (33E20) Euler-Maclaurin formula in numerical analysis (65B15)

Related Items (11)

An extension of trapezoidal type product integration rules ⋮ Convergence analysis of the generalized Euler-Maclaurin quadrature rule for solving weakly singular integral equations ⋮ Summation Formulas of Euler–Maclaurin and Abel–Plana: Old and New Results and Applications ⋮ A modified Euler–Maclaurin formula in 1D and 2D with applications in statistical physics ⋮ Asymptotic behaviors of intermediate points in the remainder of the Euler-Maclaurin formula ⋮ A new Bernoulli matrix method for solving second order linear partial differential equations with the convergence analysis ⋮ Special functions related to Dedekind-type DC-sums and their applications ⋮ A fourth order product integration rule by using the generalized Euler-Maclaurin summation formula ⋮ Numerical solution of the static beam problem by Bernoulli collocation method ⋮ On the efficient computation of large scale singular sums with applications to long-range forces in crystal lattices ⋮ Unnamed Item

Cites Work

• Asymptotic Expansions for Product Integration
• Hybrid Gauss-Trapezoidal Quadrature Rules
• A Short Proof of the Explicit Formula for Bernoulli Numbers
• Generalized Gaussian Quadrature Rules for Systems of Arbitrary Functions
• Analytical expansion and numerical approximation of the Fermi-Dirac integrals \(\mathcal F_j(x)\) of order \(j=-1/2\) and \(j=1/2\)
• Variable precision algorithm for the numerical computation of the Fermi-Dirac Function \({\mathcal F}_j(x)\) of order \(j=-3/2\)
• Über das Maximum des absoluten Betrages von \[ \frac 1{b-a}\int _a^bf(x)g(x)\,dx-\frac 1{(b-a)^2}\int _a^bf(x)\,dx\int _a^bg(x)\,dx. \ .]

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