Multiple-location matched approximation for Bessel function J0 and its derivatives

Abstract

I present an approximation of Bessel function J0(r) of the first kind for small arguments near the origin. The approximation comprises a simple cosine function that is matched with J0(r) at r=π/e. A second matching is then carried out with the standard, but slightly modified, far-field approximation for J0(r), such that zeroth, first and second derivatives are also considered. Finally, a third matching is made with the standard far-field approximation of J0 but at multiple locations, to guarantee matching all concerned derivatives. The proposed approximation is practical when nonlinear dynamics come into play, in particular in the case of nonlinear interactions that involve higher order differential equations.