Complete asymptotic expansions for the relativistic Fermi-Dirac integral

Gil, Amparo; Segura, Javier; Temme, Nico

doi:10.1016/j.amc.2021.126618

Cited by 4 publications

(1 citation statement)

References 21 publications

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“…The Fermi-Dirac integral has many applications in nuclear astrophysics and finding the concentration of electrons and holes in a semiconductor. There has been substantial research [11,12,13,14,15] on just trying to obtain theoretical and numerical approximates of the relativistic and non-relativistic Fermi-Dirac integral. In the following tables, DNNI is used to get an approximate closed-form expression for the above integrals.…”

Section: Parametric Integralsmentioning

confidence: 99%

Computing Anti-Derivatives using Deep Neural Networks

Chakraborty¹,

Gopalakrishnan²

2022

Preprint

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This paper presents a novel algorithm to obtain the closed-form anti-derivative of a function using Deep Neural Network architecture. In the past, mathematicians have developed several numerical techniques to approximate the values of definite integrals, but primitives or indefinite integrals are often non-elementary. Anti-derivatives are necessarily required when there are several parameters in an integrand and the integral obtained is a function of those parameters. There is no theoretical method that can do this for any given function. Some existing ways to get around this are primarily based on either curve fitting or infinite series approximation of the integrand, which is then integrated theoretically. Curve fitting approximations are inaccurate for highly non-linear functions and require a different approach for every problem. On the other hand, the infinite series approach does not give a closed-form solution, and their truncated forms are often inaccurate. We claim that using a single method for all integrals, our algorithm can approximate anti-derivatives to any required accuracy. We have used this algorithm to obtain the anti-derivatives of several functions, including non-elementary and oscillatory integrals. This paper also shows the applications of our method to get the closed-form expressions of elliptic integrals, Fermi-Dirac integrals, and cumulative distribution functions and decrease the computation time of the Galerkin method for differential equations.

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Section: Parametric Integralsmentioning

confidence: 99%

Computing Anti-Derivatives using Deep Neural Networks

Chakraborty¹,

Gopalakrishnan²

2022

Preprint

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Fermi-Dirac Integrals in Degenerate Regimes: Novel Asymptotic Expansion

Birrell,

Formanek,

Steinmetz

et al. 2024

Int J Theor Phys

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Computation of the confluent hypergeometric function U(a,b,x) and its derivative for positive arguments

et al. 2023

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An algorithm and a MATLAB implementation for computing the Kummer function U(a,b,x) and its derivative is given in this paper. The algorithm is efficient and accurate. Numerical tests show that the MATLAB algorithm allows the computation of the function with $\sim 10^{-14}$ ∼ 1 0 − 14 relative accuracy in the parameter region (a,b,x) ∈ (0,500) × (0,500) × (0,1000) in double-precision floating point arithmetic.

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Complete asymptotic expansions for the relativistic Fermi-Dirac integral

Cited by 4 publications

References 21 publications

Computing Anti-Derivatives using Deep Neural Networks

Computing Anti-Derivatives using Deep Neural Networks

Fermi-Dirac Integrals in Degenerate Regimes: Novel Asymptotic Expansion

Computation of the confluent hypergeometric function U(a,b,x) and its derivative for positive arguments

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