{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,12]],"date-time":"2026-02-12T16:48:29Z","timestamp":1770914909783,"version":"3.50.1"},"reference-count":22,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2020,7,16]],"date-time":"2020-07-16T00:00:00Z","timestamp":1594857600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2020,7,16]],"date-time":"2020-07-16T00:00:00Z","timestamp":1594857600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2020,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Three analytic approximations for the Bessel function <jats:inline-formula><jats:alternatives><jats:tex-math>$$J_{0}(x)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n<mml:mrow>\n<mml:msub>\n<mml:mi>J<\/mml:mi>\n<mml:mn>0<\/mml:mn>\n<\/mml:msub>\n<mml:mrow>\n<mml:mo>(<\/mml:mo>\n<mml:mi>x<\/mml:mi>\n<mml:mo>)<\/mml:mo>\n<\/mml:mrow>\n<\/mml:mrow>\n<\/mml:math><\/jats:alternatives><\/jats:inline-formula> have been determined, valid for every positive value of the variable <jats:italic>x<\/jats:italic>. The three approximations are very precise. The technique used here is based on the multipoint quasi-rational approximation method, MPQA, but here the procedure has been improved and extended. The structure of the approximation is derived considering simultaneously both the power series and asymptotic expansion of <jats:inline-formula><jats:alternatives><jats:tex-math>$$J_{0}(x)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n<mml:mrow>\n<mml:msub>\n<mml:mi>J<\/mml:mi>\n<mml:mn>0<\/mml:mn>\n<\/mml:msub>\n<mml:mrow>\n<mml:mo>(<\/mml:mo>\n<mml:mi>x<\/mml:mi>\n<mml:mo>)<\/mml:mo>\n<\/mml:mrow>\n<\/mml:mrow>\n<\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The analytic approximation is like a bridge between both expansions. The accuracy of the zeros of each approximant is even higher than the functions itself. The maximum absolute error of the best approximation is 0.00009. The maximum relative error is in the first zero and it is 0.00004.<\/jats:p>","DOI":"10.1007\/s40314-020-01238-z","type":"journal-article","created":{"date-parts":[[2020,7,16]],"date-time":"2020-07-16T06:29:16Z","timestamp":1594880956000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Analytic approximation to Bessel function $$J_{0}(x)$$"],"prefix":"10.1007","volume":"39","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9380-1277","authenticated-orcid":false,"given":"F.","family":"Maass","sequence":"first","affiliation":[]},{"given":"P.","family":"Martin","sequence":"additional","affiliation":[]},{"given":"J.","family":"Olivares","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,7,16]]},"reference":[{"issue":"1","key":"1238_CR1","first-page":"1","volume":"1","author":"M Abbaszadeh","year":"2020","unstructured":"Abbaszadeh M, Dehghan M (2020) A meshless numerical investigation based on the RBF-QR approach for elasticity problems. AUT J Math Comput 1(1):1\u201315","journal-title":"AUT J Math Comput"},{"key":"1238_CR2","first-page":"355","volume-title":"Handbook of mathematical functions, ninth printing","author":"M Abramovitz","year":"1970","unstructured":"Abramovitz M, Stegun IA (1970) Handbook of mathematical functions, ninth printing. Dover Books, New York, pp 355\u2013370"},{"issue":"1","key":"1238_CR3","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1109\/TAES.1986.310686","volume":"22","author":"NM Blachman","year":"1986","unstructured":"Blachman NM, Mousavineezhad SH (1986) Trigonometric approximation for Bessel functions. IEEE Trans Aerosp Electron Syst 22(1):2\u20137","journal-title":"IEEE Trans Aerosp Electron Syst"},{"key":"1238_CR4","unstructured":"Byron FW, Fuller RW (1992) Mathematics of classic and quantum physics. Dover Publications Inc., New York, pp 229\u2013300, 371\u2013374"},{"key":"1238_CR5","first-page":"164","volume-title":"Plasma physics and control fusion","author":"FF Chen","year":"2010","unstructured":"Chen FF (2010) Plasma physics and control fusion, vol 1, 2nd edn. Plenum Press, New York, pp 164\u2013275","edition":"2"},{"issue":"1","key":"1238_CR6","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1080\/00207160500069847","volume":"83","author":"M Dehghan","year":"2006","unstructured":"Dehghan M (2006) Solution of a partial integro-differential equation arising from viscoelasticity. Int J Comput Math 83(1):123\u2013129","journal-title":"Int J Comput Math"},{"issue":"4","key":"1238_CR7","doi-asserted-by":"crossref","first-page":"467","DOI":"10.1080\/10236190903049017","volume":"17","author":"M Dehghan","year":"2011","unstructured":"Dehghan M, Mazrooei-Sebdani R, Sedaghat H (2011) Global behaviour of the Riccati difference equation of order two. J Differ Equ Appl 17(4):467\u2013477","journal-title":"J Differ Equ Appl"},{"key":"1238_CR8","doi-asserted-by":"crossref","first-page":"140","DOI":"10.1016\/j.rinp.2017.11.019","volume":"8","author":"D Diaz","year":"2018","unstructured":"Diaz D, Martin P (2018) Approximate eigenvalues by a new technique. Application to sixth anharmonic potentials. Results Phys 8:140\u2013145","journal-title":"Results Phys"},{"key":"1238_CR9","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1007\/s00340-008-3096-1","volume":"92","author":"HT Eyyuboglu","year":"2008","unstructured":"Eyyuboglu HT, Baykal Y, Sermuthu E, Cal Y (2008) Scintillation advantages of lowest order Bessel\u2013Gaussian beams. Appl Phys B 92:229\u2013235","journal-title":"Appl Phys B"},{"key":"1238_CR10","first-page":"112","volume-title":"Classical electrodynamics","author":"JD Jackson","year":"1998","unstructured":"Jackson JD (1998) Classical electrodynamics, 3rd edn. Willey, New York, pp 112\u2013119","edition":"3"},{"key":"1238_CR11","doi-asserted-by":"crossref","first-page":"482","DOI":"10.1016\/j.ijmecsci.2014.10.019","volume":"89","author":"K Jae-Hoon","year":"2014","unstructured":"Jae-Hoon K (2014) Exact solution for stresses, strains, and displacements of a rectangular plate with an arbitrarily located circular hole subjected to in-plane bending moment. Int Mech Sci 89:482\u2013486","journal-title":"Int Mech Sci"},{"issue":"18","key":"1238_CR12","doi-asserted-by":"crossref","first-page":"6389","DOI":"10.1002\/mma.4463","volume":"40","author":"H Khosravian-Arab","year":"2017","unstructured":"Khosravian-Arab H, Dehghan M, Eslahchi MR (2017) Generalized Bessel functions: theory and their applications. Math Methods Appl Sci 40(18):6389\u20136410","journal-title":"Math Methods Appl Sci"},{"issue":"5","key":"1238_CR13","doi-asserted-by":"crossref","first-page":"957","DOI":"10.1016\/j.cpc.2010.01.008","volume":"181","author":"M Lakestani","year":"2010","unstructured":"Lakestani M, Dehghan M (2010) Numerical solution of Riccati equation using the cubic B-spline scaling functions and Chebyshev cardinal functions. Comput Phys Commun 181(5):957\u2013966","journal-title":"Comput Phys Commun"},{"key":"1238_CR14","doi-asserted-by":"crossref","first-page":"1234","DOI":"10.1016\/j.rinp.2018.01.071","volume":"8","author":"F Maass","year":"2018","unstructured":"Maass F, Martin P (2018) Precise analytic approximations for the Bessel function J1(x). Results Phys 8:1234\u20131238","journal-title":"Results Phys"},{"key":"1238_CR15","doi-asserted-by":"crossref","first-page":"8359","DOI":"10.1103\/PhysRevB.45.8359","volume":"45","author":"P Martin","year":"1992","unstructured":"Martin P, Rodriguez-Nunez JJ, Marquez JL (1992) Two dimensional hydrogen-like atoms in the presence of a magnetic field quasi-fractional approximation. Phys Rev B 45:8359\u20138362","journal-title":"Phys Rev B"},{"key":"1238_CR16","unstructured":"Martin P, Castro E, Paz JL, DeFreitas A (2009) Multi-point quasi-rational approximants in quantum chemistry, Chapter 4. In: Paz JL, Hernandez A (eds) New developments in quantum chemistry. Transworld Research Network , Kerala, pp 55\u201378 (Chapter 3)"},{"key":"1238_CR17","first-page":"130","volume":"63","author":"P Martin","year":"2017","unstructured":"Martin P, Olivares J, Sotomayor A (2017) Precise analytic approximation for the modified Bessel function I1. Rev Mex Fisica 63:130\u2013133","journal-title":"Rev Mex Fisica"},{"key":"1238_CR18","doi-asserted-by":"crossref","first-page":"1028","DOI":"10.1016\/j.rinp.2018.10.029","volume":"11","author":"P Martin","year":"2018","unstructured":"Martin P, Olivares J, Maass F, Valero E (2018) Analytic approximations for special functions, applied to the modified Bessel functions I2(x) and I2\/3(x). Results Phys 11:1028\u20131033","journal-title":"Results Phys"},{"key":"1238_CR19","doi-asserted-by":"crossref","first-page":"102986","DOI":"10.1016\/j.rinp.2020.102986","volume":"16","author":"P Martin","year":"2020","unstructured":"Martin P, Maas F, Diaz-Almeida D (2020) Accurate analytic approximation to eigenvalues of the anharmonic potential $$x^{2}+\\lambda x^8$$. Results Phys 16:102986","journal-title":"Results Phys"},{"key":"1238_CR20","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/MPE\/2006\/64132","volume":"2006","author":"M Masjed-Jamei","year":"2006","unstructured":"Masjed-Jamei M, Dehghan M (2006) Application of zero eigenvalue for solving the potential, heat, and wave equations using a sequence of special functions. Math Probl Eng 2006:1\u20139","journal-title":"Math Probl Eng"},{"key":"1238_CR21","doi-asserted-by":"publisher","first-page":"138","DOI":"10.1109\/MAP.2009.5433114","volume":"51","author":"EJ Rothwell","year":"2009","unstructured":"Rothwell EJ (2009) Exponential approximation of the Bessel functions I0,1(x), J0,1(x), Y0(x) and $$H0^{1,2}(x)$$, with Applications to Electromagnetic Scattering, Radiation, and Diffraction [EM Programmer\u2019s Notebooks]. IEEE Antennas Propag Mag 51:138\u2013147. https:\/\/doi.org\/10.1109\/MAP.2009.5433114","journal-title":"IEEE Antennas Propag Mag"},{"key":"1238_CR22","first-page":"85","volume-title":"A treatise on the theory of Bessel functions","author":"GN Watson","year":"1966","unstructured":"Watson GN (1966d) A treatise on the theory of Bessel functions, 2nd edn. Cambridge University Press, Cambridge, pp 85\u2013224","edition":"2"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-020-01238-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-020-01238-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-020-01238-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,15]],"date-time":"2021-07-15T23:28:42Z","timestamp":1626391722000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-020-01238-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,7,16]]},"references-count":22,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2020,9]]}},"alternative-id":["1238"],"URL":"https:\/\/doi.org\/10.1007\/s40314-020-01238-z","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,7,16]]},"assertion":[{"value":"4 March 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 June 2020","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 June 2020","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 July 2020","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"222"}}