On Integral Representations for the Generalised Bessel and Neumann Functions

The original Bessel differential equation that describes, among many others cylindrical acoustic or vortical waves, is a particular case of zero degree of the generalised Bessel differential equation that describes coupled acoustic–vortical waves. The solutions of the generalised Bessel differential equation can be obtained for all possible combinations of complex variable, order and degree by three alternative methods: (i) convergent power series of Frobenius–Fuchs type around the regular singularity at the origin; (ii) asymptotic expansions of Thomé normal integral type around the irregular singularity at infinity; and (iii) Laplace transform along suitable paths in the complex plane which are the focus of the present paper. This leads to the generalised Bessel, Neumann and Hankel functions of two kinds, for which are obtained: (i) power series; (ii) asymptotic expansions; and (iii/iv) representations as definite and contour integrals. For the generalised Bessel functions are obtained four integral representations: (i) as two definite integrals along the unit interval with branch-points at both ends; (ii) as integrals along tear-drop loops surrounding one branch-point each; and (iii) as an integral along a Pochhammer double-laced loop around both branch-points. For the modified generalised Neumann function are obtained two integral representations: (i) along the negative real axis joining infinity to the branch-point at origin and (ii) along a Hankel-type semi-infinite loop around the single branch-point. The generalised Hankel functions of two kinds are linear combinations of generalised Bessel and Neumann functions, and are used to describe cylindrical acoustic or vortical waves and their coupling. The acoustic–vortical waves in a cylindrical duct are illustrated as plots of generalised Bessel functions with different orders and degrees. The original Bessel functions describe purely acoustic or vortical cylindrical waves that, for large radius, are asymptotically decaying and oscillating, hence being stable. The generalised Bessel functions describe coupled acoustic–vortical waves that are asymptotically monotonic and unstable. This provides a physical interpretation for the different mathematical properties of the original and modified Bessel functions.