Three analytic approximations for the Bessel function \(J_{0}(x)\) 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 \(J_{0}(x)\). 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.

中文翻译：

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贝塞尔函数$$ J_ {0}（x）$$ J 0（x）的解析逼近

已经确定了贝塞尔函数\（J_ {0}（x）\）的三个解析近似值，它们对变量x的每个正值均有效。这三个近似值非常精确。此处使用的技术基于多点准理性逼近方法MPQA，但此处的过程已得到改进和扩展。同时考虑\（J_ {0}（x）\）的幂级数和渐近展开，得出近似结构。解析近似就像是两个展开之间的桥梁。每个近似值的零点的精度甚至高于函数本身。最佳近似的最大绝对误差为0.00009。最大相对误差在第一个零处，为0.00004。