Analytical and asymptotic evaluations of Dawson’s integral and related functions in mathematical physics

• Autorzy: Nijimbere Victor

Identyfikatory

DOI

10.1515/jaa-2019-0019

• Języki publikacji: EN

Abstrakty

EN

Dawson’s integral and related functions in mathematical physics that include the complex error function (Faddeeva’s integral), Fried-Conte (plasma dispersion) function, Jackson function, Fresnel function and Gordeyev’s integral are analytically evaluated in terms of the confluent hypergeometric function. And hence, the asymptotic expansions of these functions on the complex plane ℂ are derived by using the asymptotic expansion of the confluent hypergeometric function.

Słowa kluczowe

EN

Dawson's integral; complex error function; plasma dispersion function; Fresnel functions; Gordeyev’s integral; confluent hypergeometric function; asymptotic evaluation

PL

całka Dawsona; złożona funkcja błędu; funkcja dyspersji plazmy; funkcje Fresnela; całka Gordiejewa; funkcja hipergeometryczna konfluentna; ocena asymptotyczna

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2019
• Tom: Vol. 25, nr 2
• Strony: 179--188
• Opis fizyczny: Bibliogr. 27 poz.

Twórcy

autor

Nijimbere Victor

• School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada

Bibliografia

• [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Natl. Bureau Standards Appl. Math. Ser. 55, U.S. Government Printing Office, Washington, 1964.
• [2] S. M. Abrarov and B. M. Quine, Efficient algorithmic implementation of the Voigt/complex error function based on exponential series approximation, Appl. Math. Comput. 218 (2011), no. 5, 1894-1902.
• [3] S. M. Abrarov and B. M. Quine, Accurate approximations for the complex error function with small imaginary argument, J. Math. Res. 7 (2015), 44-53.
• [4] W. Baumjohann and R. A. Treumann, Basic Space Plasma Physics, Imperial College Press, Lodon, 1997.
• [5] H. Borchert, D. V. Talapin, N. Gaponik, C. McGniley, S. Adam, A. Lobo, T. Möller and H. Weller, Relations between the photoluminescence efficiency of CdTe nanocrystals and their surface properties revealed by synchrotron XPS, J. Phys. Chem. B 107 (2003), 9662-9668.
• [6] I. Charpentier and J. Gustedt, Arbogast: Higher order automatic differentiation for special functions with Modular C, Optim. Methods Softw. 33 (2018), no. 4-6, 963-987.
• [7] W. J. Cody, K. A. Paciorek and H. C. Thacher, Jr., Chebyshev approximations for Dawson’s integral, Math. Comp. 24 (1970), 171-178.
• [8] V. N. Fadeeva and N. M. Terent’ev, Tables of Values of the Probabilistic Integral for Complex Arguments, Moscow State Publishing House for Technical Theoritical Literature, Moscow, 1954.
• [9] V. N. Fadeeva and N. M. Terent’ev, Tables of the Probabilistic Integral w(z) = e−z2 (1 + 2i √π ∫z 0 et2 dt) for Complex Arguments, Oxford Pargamon Press, Oxford, 1961.
• [10] B. D. Fried and S. D. Conte, The Plasma Dispersion Function, Academic Press, New York, 1961.
• [11] W. Gautschi, Efficient computation of the complex error function, SIAM J. Numer. Anal. 7 (1970), 187-198.
• [12] G. V. Gordeyev, Plasma oscillations in a magnetic field, Sov. Phys. JETP 6 (1952), 660-669.
• [13] E. A. Jackson, Drift instabilities in a Maxwellian plasma, Phys. Fluids 3 (1960), 786-792.
• [14] E. A. Marchisotto and G.-A. Zakeri, An invitation to integration in finite terms, College Math. J. 25 (1994), 295-308.
• [15] J. H. McCabe, A continued fraction expansion, with a truncation error estimate, for Dawson’s integral, Math. Comp. 28 (1974), 811-816.
• [16] S. J. McKenna, A method of computing the complex probability function and other related functions over the whole complex plane, Astrophys. Space Sci. 107 (1984), 71-83.
• [17] A. B. Mikhailovskiy, Theory of plasma instabilities (in Russian), Russia Atomizdat, 1975.
• [18] V. Nijimbere, Evaluation of the non-elementary integral ∫ eλxα dx, α ≥ 2, and other related integrals, Ural Math. J. 3 (2017), no. 2, 130-142.
• [19] R. B. Paris, The asymptotic expansion of Gordeyev’s integral, Z. Angew. Math. Phys. 49 (1998), no. 2, 322-338.
• [20] G. P. M. Poppe and C. M. J. Wijers, More efficient computation of the complex error function, ACM Trans. Math. Software 16 (1990), no. 1, 38-46.
• [21] M. Rosenlicht, Integration in finite terms, Amer. Math. Monthly 79 (1972), 963-972.
• [22] F. Schreier, The Voigt and complex error functions: A comparison of computational methods, J. Quant. Spectrosc. Radiat. Transfer 48 (1992), 743-762.
• [23] A. G. Sitenko, Fluctuations and Nonlinear Wave Interactions in Plasmas. Translated from the Russian by O. D. Kocherga, International Series in Natural Philosophy 107, Pergamon Press, New York, 1982.
• [24] T. H. Stix, The Theory of Plasma Waves, McGraw-Hill, New York, 1962.
• [25] J. A. C. Weideman, Computation of the complex error function, SIAM J. Numer. Anal. 31 (1994), no. 5, 1497-1518.
• [26] M. R. Zaghloul and A. N. Ali, Algorithm 916: Computing the Faddeyeva and Voigt functions, ACM Trans. Math. Software 38 (2011), no. 2, Paper No. 15.
• [27] NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).