Physical Science: New Insights and Developments Vol. 3

The power series of \(J_1\)(x) is well known and its convergence radius is infinite. In this work, an analytic approximation for \(J_1\)(x) has been found, which is simple and precise, and good for most of the applications of these functions in Physics. Two techniques have been used here, and the simplest approximant is a function of four parameters. The technique used here resembles a little the Pade method, since rational functions are used, but now this type of function is combined in an efficient way with elementary functions. Furthermore, series power and asymptotic expansions are used simultaneously, as in the Multipoint Quasi-rational Approximation MPQA method. However, here important improvements have been introduced. Though the form of the approximate is built considering the above two expansions, however the parameters of the approximations are determinates by two methods, one similar to the minimum square error method and the other using the coefficients of two expansions, power and asymptotic. The resulting approximations are very simple yet achieve high accuracy, sufficient for most physical applications of \(J_1\)(x). Better results are obtained with the first procedure.