Accurate Analytical Approximation for the Bessel Function J2(x)
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2024
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We obtain an accurate analytic approximation for the Bessel function š½2(š„)
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 š„=0
to š„=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 š½2(š„)
and the approximated function š½Ė2(š„) occur are also provided, exhibiting very small relative errors. The largest relative error is for the second zero, with šrel=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.
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Bessel function, MPQA