A General Analytic Approximation Technique for the Modified Bessel Functions with Fractional Order I v (x) , Applied to I 1/6 (x) and I 1/7 (x)

Abstract

A general technique to obtain simple analytic approximations for the first kind of modified Bessel functions. The general procedure is shown, and the parameter determination is explained through the applications to this particular case I 1 / 6 ( x ) and I 1 / 7 ( x ) . In this way, it shows how to apply the technique to any particular order ν , in order to obtain an approximation valid for any positive value of the variable x . In the present method power series and asymptotic expansion are simultaneously used. The technique is an extension of the multipoint quasirational approximation method, MPQA. The main idea is to look for a bridge function between the power and asymptotic expansion of the I 1 / 6 ( x ) , and similar procedure for I 1 / 7 ( x ) . To perform this, rational functions are combined with hyperbolic ones and fractional powers. The number of parameters to be determined for each case is four. The maximum relative errors are 0 . 0049 for ν = 1 / 6 , and 0 . 0047 for ν = 7 . However, these relative errors decrease outside of the small region of the variables, wherein the maximum relative errors are reached. There is a clear advantage of this procedure compared with any other ones.