On Generalised Hankel Functions and a Bifurcation of Their Asymptotic Expansion

Abstract

The generalised Bessel differential equation has an extra parameter relative to the original Bessel equation and its asymptotic solutions are the generalised Hankel functions of two kinds distinct from the original Hankel functions. The generalised Bessel differential equation of order ν and degree μ reduces to the original Bessel differential equation of order ν for zero degree, μ = 0. In both cases the differential equations have a regular singularity near the origin and the the point at infinity is the other singularity. The point at infinity is an irregular singularity of different degree, namely one for the original and two for the generalised Bessel differential equation. It follows that in the limit of degree being equal to zero the generalised Hankel functions do not converge to the original ones. The implication is that the generalised Bessel differential equation has a Hopf-type bifurcation for the asymptotic solution. In the case of a real variable and parameters the asymptotic solution is: (i) oscillatory when the degree of generalised Hankel function is zero (corresponding in this case to original Hankel functions); (ii) diverging hence unstable for the generalised Hankel functions with positive degree; (iii) decaying hence stable for the generalised Hankel functions with negative degree.