Quadrature formulas with exponential convergence and calculation of the Fermi–Dirac integrals
- Authors: Kalitkin N.N.1, Kolganov S.A.2
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Affiliations:
- Keldysh Institute of Applied Mathematics
- National Research University of Electronic Technology (MIET)
- Issue: Vol 95, No 2 (2017)
- Pages: 157-160
- Section: Mathematics
- URL: https://journals.rcsi.science/1064-5624/article/view/224929
- DOI: https://doi.org/10.1134/S1064562417020156
- ID: 224929
Cite item
Abstract
A class of functions for which the trapezoidal rule has superpower convergence is described: these are infinitely differentiable functions all of whose odd derivatives take equal values at the left and right endpoints of the integration interval. An heuristic law is revealed; namely, the convergence exponentially depends on the number of nodes, and the exponent equals the ratio of the length of integration interval to the distance from this interval to the nearest pole of the integrand. On the basis of the obtained formulas, a method for calculating the Fermi–Dirac integrals of half-integer order is proposed, which is substantially more economical than all known computational methods. As a byproduct, an asymptotics of the Bernoulli numbers is found.
About the authors
N. N. Kalitkin
Keldysh Institute of Applied Mathematics
Author for correspondence.
Email: kalitkin@imamod.ru
Russian Federation, Moscow, 125047
S. A. Kolganov
National Research University of Electronic Technology (MIET)
Email: kalitkin@imamod.ru
Russian Federation, Zelenograd, Moscow, 124498