Gaussian integral by Taylor series and applications

Lázaro Lima de Sales; Jonatas Arizilanio Silva; Eliângela Paulino Bento de Souza; Hidalyn Theodory Clemente Mattos de Souza; Antonio Diego Silva Farias; Otávio Paulino Lavor

doi:10.35819/remat2021v7i2id4330

Authors

Lázaro Lima de Sales Universidade do Estado do Rio Grande do Norte (UERN), Departamento de Física, Mossoró, RN, Brasil https://orcid.org/0000-0002-5352-6642
Jonatas Arizilanio Silva Universidade Federal Rural do Semi-Árido (UFERSA), Departamento de Ciência e Tecnologia (DCT), Caraúbas, RN, Brasil https://orcid.org/0000-0003-2246-1660
Eliângela Paulino Bento de Souza Universidade Federal da Paraíba (UFPB), Departamento de Física, João Pessoa, PB, Brasil https://orcid.org/0000-0002-4466-2802
Hidalyn Theodory Clemente Mattos de Souza Universidade Federal Rural do Semi-Árido (UFERSA), Departamento de Ciências Exatas e Naturais (DECEN), Pau dos Ferros, RN, Brasil https://orcid.org/0000-0002-7874-8188
Antonio Diego Silva Farias Universidade Federal Rural do Semi-Árido (UFERSA), Departamento de Ciências Exatas e Naturais (DECEN), Pau dos Ferros, RN, Brasil https://orcid.org/0000-0002-1222-7013
Otávio Paulino Lavor Universidade Federal Rural do Semi-Árido (UFERSA), Departamento de Ciências Exatas e Naturais (DECEN), Pau dos Ferros, RN, Brasil http://orcid.org/0000-0001-5237-3392

DOI:

https://doi.org/10.35819/remat2021v7i2id4330

Keywords:

Gaussian Integral, Special Functions, Fractional Derivative

Abstract

In this paper, we present a solution for a specific Gaussian integral. Introducing a parameter that depends on a n index, we found out a general solution inspired by the Taylor series of a simple function. We demonstrated that this parameter represents the expansion coefficients of this function, a very interesting and new result. We also introduced some Theorems that are proved by mathematical induction. As a test for the solution presented here, we investigated a non-extensive version for the particle number density in Tsallis framework, which enabled us to evaluate the functionality of the method. Besides, solutions for a certain class of the gamma and factorial functions are derived. Moreover, we presented a simple application in fractional calculus. In conclusion, we believe in the relevance of this work because it presents a solution for the Gaussian integral from an unprecedented perspective.

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Author Biographies

Lázaro Lima de Sales, Universidade do Estado do Rio Grande do Norte (UERN), Departamento de Física, Mossoró, RN, Brasil

http://lattes.cnpq.br/7727890574597438
Jonatas Arizilanio Silva, Universidade Federal Rural do Semi-Árido (UFERSA), Departamento de Ciência e Tecnologia (DCT), Caraúbas, RN, Brasil

http://lattes.cnpq.br/0385112916465731
Eliângela Paulino Bento de Souza, Universidade Federal da Paraíba (UFPB), Departamento de Física, João Pessoa, PB, Brasil

http://lattes.cnpq.br/5621970048397580
Hidalyn Theodory Clemente Mattos de Souza, Universidade Federal Rural do Semi-Árido (UFERSA), Departamento de Ciências Exatas e Naturais (DECEN), Pau dos Ferros, RN, Brasil

http://lattes.cnpq.br/1340176705421593
Antonio Diego Silva Farias, Universidade Federal Rural do Semi-Árido (UFERSA), Departamento de Ciências Exatas e Naturais (DECEN), Pau dos Ferros, RN, Brasil

http://lattes.cnpq.br/4450585340269458
Otávio Paulino Lavor, Universidade Federal Rural do Semi-Árido (UFERSA), Departamento de Ciências Exatas e Naturais (DECEN), Pau dos Ferros, RN, Brasil

http://lattes.cnpq.br/1857806253382088

References

ARFKEN, G. B.; WEBER, H. J. Mathematical Methods for Physicists. New York: Academic Press, 2005.

BOAS, M. L. Mathematical methods in the physical sciences. New Jersey: John Wiley & Sons, 2006.

CONRAD, K. T. The Gaussian Integral, 2013. Available in: https://www.semanticscholar.org/paper/THE-GAUSSIAN-INTEGRAL-Conrad/4687538f80e333c175691d627dc1254eef3605f8. Access in: 2020.

DAVIS, P. J. Leonhard Euler's Integral: A Historical Profile of the Gamma Function: In Memoriam: Milton Abramowitz. The American Mathematical Monthly, v. 66, p. 847-869, 1959.

GREINER, W. Quantum Mechanics: An Introduction. Berlin: Springer, 1990.

GREINER, W. Thermodynamics and Statistical Mechanics. Berlin: Springer, 1995.

GRONAU, D. Why is the gamma function so as it is? Teaching Mathematics and Cumputer Science, v. 1, p. 43-53, 2003.

HERNANDEZ, S. M. Termodinàmica i Mecànica estadìstica. London: Lulu, 2015.

LAPLACE, P. S. Théorie Analytiques des Probabilités. Paris: Courcier, 1820.

OLIVEIRA, E. C.; MACHADO, J. A. T. A review of definitions for fractional derivatives and integral. Mathematical Problems in Engineering, v. 2014, p. 1-6, 2014.

PATHRIA, R. K. Statistical Mechanics. Oxford: Butterworth-Heinemann, 1996.

PESSAH, M. E.; TORRES, D. F.; VUCETICH, H. Statistical mechanics and the description of the early universe. (I). Foundations for a slightly non-extensive cosmology. Physica A, v. 297, p. 164-200, 2001.

RILEY, K. F.; HOBSON, M. P.; BENCE , S. J. Mathematical Methods for Physics and Engineering. Cambridge: Cambridge University Press, 2006.

SAKURAI, J. J. Modern Quantum Mechanics. New York: Addison-Wesley, 1985.

SALINAS, S. R. A. Introduction to Statistical Physics. New York: Springer, 2001.

SHEN, K. M.; ZHANG, B. W.; WANG, E. K. Generalized ensemble theory with non-extensive statistics. Physica A, v. 487, p. 215-224, 2017.

SPIEGEL, M. R. ; SCHILLER, J. ; SRINIVASAN, R. A. Schaum's Outline of Probability and Statistics. New York: McGraw-Hill, 2001.

STAHL, S. The Evolution of the Normal Distribution. Mathematics Magazine, v. 79, n. 2, p. 96-113, 2006.

STIGLER, S. M. Laplace's 1774 memoir on inverse probability. Statistical Science, v. 1, p. 359-378, 1986.

STURM, J. K. F. Cours d'Analyse de l’école polytechnique. Paris: Mallet-Bachelier, 1857.

TSALLIS, C. Possible generalization of Boltzmann-Gibbs statistics. Journal of Statistical Physics, v. 52, p. 479-487, 1988.

WEISSTEIN, E. W. Gaussian integral. From MathWorld-A Wolfram Web Resource. Available in: http://mathworld.wolfram.com/GaussianIntegral.html. Access in: 2020.

WEISSTEIN, E. W. Hypergeometric Function. From MathWorld-A Wolfram Web Resource. Available in: https://mathworld.wolfram.com/HypergeometricFunction.html. Access in: 2020.