An Improved Analytical Approximation of the Bessel Function J2(x)

In this paper, we derive an analytical and explicit approximation for the Bessel function J2(x) for positive real x with a maximum absolute error of approximately 0.004, which refines some new published analytic approximation. The absolute errors for large x decrease logarithmically, according to our analysis of these errors in the interval [0,103]. The power series and the asymptotic series are used in the process to produce this analytic approximation. Additionally, extremely minor relative errors are shown for the values of x at which the zeros of our approximation function and the function J2(x) occur. The first positive zero has the biggest relative error equal to 0.00219574. The relative errors then steadily drop, reaching 9.13302×10−7 for the eleventh zero.