Analytic approximation to Bessel function $$J_{0}(x)$$

Abstract

AbstractThree analytic approximations for the Bessel function $$J_{0}(x)$$ J 0 ( 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 $$J_{0}(x)$$ J 0 ( 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.