Analytic approximation to Bessel function $$J_{0}(x)$$ J 0 ( x )
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Analytic approximation to Bessel function J0 (x) F. Maass1
· P. Martin1 · J. Olivares2
Received: 4 March 2020 / Revised: 15 June 2020 / Accepted: 25 June 2020 © The Author(s) 2020
Abstract Three analytic approximations for the Bessel function J0 (x) have been determined, valid for every positive value of the variable x. 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 J0 (x). 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. Keywords Bessel functions · Quasirational approximation · Asymptotic treatment Mathematics Subject Classification 33C10 · 41A20 · 26A06 · 26A99
1 Introduction The Bessel function J0 (x) is present in a lot of applications like electrodynamics (Jackson 1998; Blachman and Mousavineezhad 1986; Rothwell 2009), mechanics (Kang 2014), diffusion in cylinder and waves in kinetic theory in plasma physics (Chen 2010), generalized Bessel functions are investigated in Khosravian-Arab et al. (2017) and others relevant application for especial function are in Masjed-Jamei and Dehghan (2006), Dehghan et al. (2011), Lakestani and Dehghan (2010), Dehghan (2006), Abbaszadeh and Dehghan (2020), Watson (1966), Byron and Fuller (1992) and Eyyuboglu et al. (2008). There are several kinds
Communicated by Antonio José Silva Neto.
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F. Maass [email protected] P. Martin [email protected] J. Olivares [email protected]
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Departamento de Fisica, Universidad de Antofagasta, Av. Angamos 601, Antofagasta, Chile
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Departamento de Matemáticas, Universidad de Antofagasta, Av. Angamos 601, Antofagasta, Chile 0123456789().: V,-vol
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of approximations including polynomials ones, each one limited to a given interval of the variable (Abramovitz and Stegun 1970). The series expansion of this function is entire, but for large values of the variable you need a lot of terms to get good accuracy, and its calculation becomes cumbersome. An analytic and very accurate approximation is presented here, which is good for every positive value of the variable, including complex value, and very useful in the applications. Furthermore, it is simple to calculate, and it can be differentiated and integrated as the exact one. This approximation is determined using only eight parameters, and the way to obtain is by a peculiar method, which uses simultaneously the power and asymptotic expansion of the function. In this way, a bridge function between both expansions is built and the result is very good. That is, the accuracy is very high for both the function and its zeros, which
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