Accurate Analytical Approximation for the Bessel Function J2(x)

Martin, Pablo; Ramos-Andrade, Juan Pablo; Caro-Pérez, Fabián; Lastra, Freddy

doi:10.3390/mca29040063

Open AccessArticle

Accurate Analytical Approximation for the Bessel Function J₂(x)

Departamento de Física, Facultad de Ciencias Básicas, Universidad de Antofagasta, Casilla 170, Av. Angamos 601, Antofagasta 1240000, Chile

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Math. Comput. Appl. 2024, 29(4), 63; https://doi.org/10.3390/mca29040063

Submission received: 4 June 2024 / Revised: 1 August 2024 / Accepted: 7 August 2024 / Published: 9 August 2024

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Abstract

We obtain an accurate analytic approximation for the Bessel function

J_{2} (x)

using an improved multipoint quasirational approximation technique (MPQA). This new approximation is valid for all real values of the variable x, with a maximum absolute error of approximately 0.009. These errors have been analyzed in the interval from

x = 0

to

x = 1000

, and we have found that the absolute errors for large x decrease logarithmically. The values of x at which the zeros of the exact function

J_{2} (x)

and the approximated function

{\tilde{J}}_{2} (x)

occur are also provided, exhibiting very small relative errors. The largest relative error is for the second zero, with

ε_{rel} = 0.0004

, and the relative errors continuously decrease, reaching

0.0001

for the eleventh zero. The procedure to obtain this analytic approximation involves constructing a bridge function that connects the power series with the asymptotic approximation. This is achieved by using rational functions combined with other elementary functions, such as trigonometric and fractional power functions.

Keywords: Bessel function; MPQA

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MDPI and ACS Style

Martin, P.; Ramos-Andrade, J.P.; Caro-Pérez, F.; Lastra, F. Accurate Analytical Approximation for the Bessel Function J₂(x). Math. Comput. Appl. 2024, 29, 63. https://doi.org/10.3390/mca29040063

AMA Style

Martin P, Ramos-Andrade JP, Caro-Pérez F, Lastra F. Accurate Analytical Approximation for the Bessel Function J₂(x). Mathematical and Computational Applications. 2024; 29(4):63. https://doi.org/10.3390/mca29040063

Chicago/Turabian Style

Martin, Pablo, Juan Pablo Ramos-Andrade, Fabián Caro-Pérez, and Freddy Lastra. 2024. "Accurate Analytical Approximation for the Bessel Function J₂(x)" Mathematical and Computational Applications 29, no. 4: 63. https://doi.org/10.3390/mca29040063

APA Style

Martin, P., Ramos-Andrade, J. P., Caro-Pérez, F., & Lastra, F. (2024). Accurate Analytical Approximation for the Bessel Function J₂(x). Mathematical and Computational Applications, 29(4), 63. https://doi.org/10.3390/mca29040063

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Accurate Analytical Approximation for the Bessel Function J₂(x)

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