On Asymptotic Expansions for Generalised and Modified Neumann Functions

It is shown that the generalised Bessel and confluent hypergeometric differential equations can be transformed into each other and that their solutions can be related. The confluent hypergeometric function of the first kind corresponds to the generalised Bessel function; the confluent hypergeometric function of the second kind corresponds to a modified generalised Neumann function, that is a linear combination of generalised Bessel and Neumann functions. The generalised Hankel functions of the first kind appear in the asymptotic expansion of the modified generalised Neumann function. The generalised Hankel function of the second kind is needed in the asymptotic expansions for generalised Bessel and (unmodified) generalised Neumann functions. As an application is considered the wave propagation in a cylindrical duct with (a) uniform axial flow and (b) swirl with constant angular velocity. Whereas the original Bessel differential equation applies to case (a), it is shown that the inclusion of (b) swirl leads to the generalised Bessel differential equation. The plots of generalised compared with original Bessel functions show the differences between acoustic and acoustic-vortical waves, including radial and temporal instability of the latter.