An Analytic Approximation for the Bessel Function J ν ( x ) for −1/2 < ν < 3/2

We found analytic approximations for the Bessel function of the first kind J ν ( x ) , valid for any real value of x and any value of ν in the interval (−1/2, 3/2). The present approximation is exact for ν = − 1 / 2 , ν = 1 / 2 , and ν = 3 / 2 , where an exact function for each case is well known. The maximum absolute errors for ν near these peculiar values are very small. Throughout the interval, the absolute values remain below 0.05. The structure of the approximate function is defined considering the corresponding power series and asymptotic expansions, and they are quotients of three polynomials of the second degree combined with trigonometrical functions and fractional powers. This is, in some way, the Multipoint Quasi-rational Approximation (MPQA) technique, but now only two variables are considered, x and ν , which is novel, since in all previous publications only the variable x was considered and ν was given. Furthermore, in the case of J − 1 / 2 ( x ) , J 1 / 2 ( x ) , and J 3 / 2 ( x ) , the corresponding exact function was also a condition to be considered and fulfilled. It is important to point out that the zeros of the exact functions and the approximate ones are also almost coincident with small relative errors. Finally, the approximation presented here has the property of preservation of symmetry for ν > 0 , i.e., when there is a sign change in the variable x , the corresponding change agrees with a similar change in the power series of the exact function.