Potential of High Accuracy Approximation for the Modified Bessel Function of Fractional Order \(I_{1/3}(x)\), Using MPQA Method with Hyperbolic Functions

Abstract

The modified Bessel functions of fractional order \(I_{1/3}(x)\) has been approximated by an analytic function containing rational and hyperbolic functions. The Bessel functions the order 1/3 are very important, because its connection with the Airy functions. A technique using both power series and assymptotic expansion has been used. An approximation has been found for the modified Bessel function \(I_{1/3}(x)\). The accuracy of the approximation is very high using only three parameters., and the largest relative error is smaller than 0,004.