Precise and fast computation of Fermi-Dirac integral of integer and half integer order by piecewise minimax rational approximation

DOI10.1016/j.amc.2015.03.009zbMath1464.65024OpenAlexW1965569373MaRDI QIDQ1636895

Toshio Fukushima

Publication date: 7 June 2018

Published in: Applied Mathematics and Computation (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/j.amc.2015.03.009

zbMATH Keywords

Chebyshev series expansionFermi-Dirac integralSommerfeld expansionMaclaurin series expansionminimax rational approximation

Mathematics Subject Classification ID

Computation of special functions and constants, construction of tables (65D20) Statistical mechanics of solids (82D20) Other functions defined by series and integrals (33E20) Approximation algorithms (68W25) Numerical integration (65D30)

Related Items (4)

Precise and fast computation of inverse Fermi-Dirac integral of order 1/2 by minimax rational function approximation ⋮ InvFD, an OCTAVE routine for the numerical inversion of the Fermi-Dirac integral ⋮ Precise and fast computation of generalized Fermi-Dirac integral by parameter polynomial approximation ⋮ Complete asymptotic expansions for the relativistic Fermi-Dirac integral

Uses Software

• Mathematica
• Algorithm 745
• DLMF

Cites Work

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• Computation of a general integral of Fermi-Dirac distribution by McDougall-Stoner method
• Numerical inversion of a general incomplete elliptic integral
• Analytical computation of generalized Fermi-Dirac integrals by truncated Sommerfeld expansions
• Some remarks on multivariate Chebyshev polynomials
• Recursive computation of derivatives of elliptic functions and of incomplete elliptic integrals
• Double exponential formulas for numerical integration
• Gaussian integration with rescaling of abscissas and weights
• Complete asymptotic expansions of the Fermi–Dirac integrals Fp(η)=1/Γ(p+1)∫∞[εp/(1+eε−η)dε]
• The Fermi-Dirac integrals $$\mathcal{F}_p (\eta ) = (p!)^{ - 1} \int\limits_0^\infty {\varepsilon ^p (e^{\varepsilon - \eta } + 1} )^{ - 1} d\varepsilon $$
• Algorithm 779: Fermi-Dirac functions of order -1/2, 1/2, 3/2, 5/2
• Algorithm 745: computation of the complete and incomplete Fermi-Dirac integral
• Generalizations of Horner's Rule for Polynomial Evaluation
• Rational Chebyshev approximations for Fermi-Dirac integrals of orders -1/2, 1/2 and 3/2
• Fermi-Dirac functions of integral order

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