Abstract
In this paper, we establish an asymptotic expansion for the Euler–Mascheroni constant. Based on this expansion, we establish a two-sided inequality and a continued fraction approximation for the Euler–Mascheroni constant.
Similar content being viewed by others
References
Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs\(,\) and Mathematical Tables, Applied Mathematics Series 55. Ninth Printing, National Bureau of Standards, Washington, D.C. (1972)
Allasia, G., Giordano, C., Pećarić, J.: Inequalities for the gamma function relating to asymptotic expansions. Math. Inequal. Appl. 5(3), 543–555 (2002)
Alzer, H.: On some inequalities for the gamma and psi functions. Math. Comput. 66, 373–389 (1997)
Chen, C.-P.: Inequalities for the Lugo and Euler-Mascheroni constants. Appl. Math. Lett. 25(4), 787–792 (2012)
Chen, C.-P.: Approximation formulas and inequalities for the Euler-Mascheroni constant, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 115(2), Article 56 (2021). https://doi.org/10.1007/s13398-021-00999-4
Chen, C.-P., Choi, J.: Inequalities and asymptotic expansions for the constants of Landau and Lebesgue. Appl. Math. Comput. 248, 610–624 (2014)
Chen, C.-P., Mortici, C.: New sequence converging towards the Euler-Mascheroni constant. Comput. Math. Appl. 64, 391–398 (2012)
Chen, C.-P., Srivastava, H.M.: New representations for the Lugo and Euler-Mascheroni constants. Appl. Math. Lett. 24(7), 1239–1244 (2011)
Chen, C.-P., Srivastava, H.M.: New representations for the Lugo and Euler-Mascheroni constants, II. Appl. Math. Lett. 25(3), 333–338 (2012)
Chen, C.-P., Srivastava, H.M., Li, L., Manyama, S.: Inequalities and monotonicity properties for the psi (or digamma) function and estimates for the Euler-Mascheroni constant. Integral Transforms Spec. Funct. 22, 681–693 (2011)
Chen, C.-P., Srivastava, H.M., Wang, Q.: A method to construct continued fraction approximations and its applications, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 115(3), Article 97 (2021)
Chen, C.-P., Wang, Q.: Asymptotic expansions and continued fraction approximations for the harmonic numbers. Appl. Anal. Discrete Math. 13(2), 569–582 (2019)
Choi, J.: Some mathematical constants. Appl. Math. Comput. 187, 122–140 (2007)
Choi, J., Srivastava, H.M.: Integral representations for the Euler-Mascheroni constant \(\gamma \). Integral Transforms Spec. Funct. 21(9), 675–690 (2010)
Dence, T.P., Dence, J.B.: A survey of Euler’s constant. Math. Mag. 82, 255–265 (2009)
DeTemple, D.W.: The non-integer property of sums of reciprocals of consecutive integers. Math. Gaz. 75, 193–194 (1991)
DeTemple, D.W.: A quicker convergence to Euler’s constant. Am. Math. Mon. 100, 468–470 (1993)
Gavrea, I., Ivan, M.: Optimal rate of convergence for sequences of a prescribed form. J. Math. Anal. Appl. 402, 35–43 (2013)
Havil, J.: Gamma: exploring Euler’s constant. Princeton University Press, Princeton (2003)
Lagarias, J.C.: Euler’s constant: Euler’s work and modern developments. Bull. Am. Math. Soc. 50(4), 527–628 (2013)
Lin, L.: Asymptotic formulas associated with psi function with applications. J. Math. Anal. Appl. 405, 52–56 (2013)
Mortici, C.: On new sequences converging towards the Euler-Mascheroni constant. Comput. Math. Appl. 59, 2610–2614 (2010)
Negoi, T.: A faster convergence to the constant of Euler. Gazeta Matematică, seria A 15, 111–113 (1997). ((in Romanian))
Rippon, P.J.: Convergence with pictures. Am. Math. Mon. 93, 476–478 (1986)
Sondow, J.: Criteria for irrationality of Euler’s constant. Proc. Am. Math. Soc. 131(11), 3335–3345 (2003)
Srivastava, H.M.: A survey of some recent developments on higher transcendental functions of analytic number theory and applied mathematics. Symmetry 13 2294, 1–22 (2021)
Yang, S.: On an open problem of Chen and Mortici concerning the Euler-Mascheroni constant. J. Math. Anal. Appl. 396, 689–693 (2012)
Young, R.M.: Euler’s constant. Math. Gaz. 75, 187–190 (1991)
Acknowledgements
This work was the Fundamental Research Funds for the Universities of Henan Province (NSFRF210446).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflicts of interest.
Additional information
Communicated by Saeid Maghsoudi.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Han, XF., Chen, CP. Approximations to the Euler–Mascheroni Constant. Bull. Iran. Math. Soc. 49, 76 (2023). https://doi.org/10.1007/s41980-023-00820-5
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s41980-023-00820-5