Accurate analytic approximation to the Modified Bessel function of Second Kind K 0(x)

Abstract

Two analytic approximations have been determined for the modified Bessel functions of second kind K0(x), good for either positive or negative values of x. The method uses both power series and asymptotic expansions, and they approximate functions like a bridge between both expansions. Here, an improvement has been also introduced to the multipoint quasirational approximation by combining equations and numerical computations. This is done in such a way that the maximum relative errors become much smaller than those obtained with the usual multipoint quasirational technique. The maximum relative error of the best approximation here found is 0.0004, using a small number of parameters. The structure of the approximations presented here, includes two rational functions, one hyperbolic function and one fractional power. The advantages of the new approximations compared with others previously published are also shown. It seems that the accuracy of the new approximation will be enough for most of the applications of K0(x)in Physics and Engineering.

Keywords

• Bessel functions
• Multipoint approximations
• Quasirational approximations
• Asymptotic treatment