{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T00:56:04Z","timestamp":1760057764194,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,2,22]],"date-time":"2025-02-22T00:00:00Z","timestamp":1740182400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we derive an analytical and explicit approximation for the Bessel function J2(x) for positive real x with a maximum absolute error of approximately 0.004, which refines some new published analytic approximation. The absolute errors for large x decrease logarithmically, according to our analysis of these errors in the interval [0,103]. The power series and the asymptotic series are used in the process to produce this analytic approximation. Additionally, extremely minor relative errors are shown for the values of x at which the zeros of our approximation function and the function J2(x) occur. The first positive zero has the biggest relative error equal to 0.00219574. The relative errors then steadily drop, reaching 9.13302\u00d710\u22127 for the eleventh zero.<\/jats:p>","DOI":"10.3390\/axioms14030157","type":"journal-article","created":{"date-parts":[[2025,2,24]],"date-time":"2025-02-24T05:36:47Z","timestamp":1740375407000},"page":"157","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["An Improved Analytical Approximation of the Bessel Function J2(x)"],"prefix":"10.3390","volume":"14","author":[{"given":"Mansour","family":"Mahmoud","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8653-6241","authenticated-orcid":false,"given":"Hanan","family":"Almuashi","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Faculty of Science, University of Jeddah, P.O. Box 80327, Jeddah 21589, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2025,2,22]]},"reference":[{"key":"ref_1","unstructured":"Watson, G.N. (1922). A Treatise on the Theory of Bessel Functions, Cambridge University Press."},{"key":"ref_2","unstructured":"Abramowitz, M., and Stegun, I.A. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications."},{"key":"ref_3","unstructured":"Bowman, F. (1958). Introduction to Bessel Functions, Dover Publications."},{"key":"ref_4","unstructured":"Strutt, J.W. (1945). The Theory of Sound, Dover Publications."},{"key":"ref_5","unstructured":"Olver, F.W.J. (1974). Asymptotics and Special Functions, AK Peters\/CRC Press."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/0771-050X(81)90040-1","article-title":"Approximation of the Bessel Function J\u03bd(x) by Numerical Integration","volume":"7","author":"Baratella","year":"1981","journal-title":"J. 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Appl., 29.","DOI":"10.3390\/mca29040063"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/3\/157\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:40:23Z","timestamp":1760028023000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/3\/157"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,22]]},"references-count":22,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2025,3]]}},"alternative-id":["axioms14030157"],"URL":"https:\/\/doi.org\/10.3390\/axioms14030157","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,2,22]]}}}