A simple approximation for the modified Bessel function of zero order I0(x)

Abstract

New efficient analytic approximations have been found for the modified zero-order Bessel functions I0(x). The method, used herein, improves prior techniques, such as multipoint quasi-rational approximations, MPQA. The present work is also an improvement of previous works, in the sense that the form of the approach is more efficient, that is, a smaller relative error is obtained using a lower number of parameters. As in the Pade method rational functions are used with coefficients determined from the powers series, but now also asymptotic expansions are also used simultaneously with that series, and consequently the rational functions have to be combined with elementary functions, in such a way, that the approach is a bridge between both expansions. The approximation now found is a rational function combined with a hyperbolic function. Only three parameters are needed to obtain a relative error smaller than 0.6 percent.