Abstract
The original Bessel differential equation that describes, among many others, cylindrical acoustic or vortical waves, is a particular case of zero degree of the generalized Bessel differential equation that describes coupled acoustic-vortical waves. The solutions of the generalized Bessel differential equation are obtained for all possible combinations of the two complex parameters, order and degree, and finite complex variable, as Frobenius-Fuchs series around the regular singularity at the origin; the series converge in the whole complex plane of the variable, except for the point-at-infinity, that is, the only other singularity of the differential equation. The regular integral solutions of the first and second kinds lead, respectively, to the generalized Bessel and Neumann functions; these reduce to the original Bessel and Neumann functions for zero degree and have alternative expressions for nonzero degree.
Highlights
The Bessel differential equation was first considered in connexion with the oscillations of a heavy chain [1] and vibrations of a circular membrane [2] and has had since [3] a vast number of applications supported by an extensive theory [4]
] = n, the general integral (18b) fails (18a), and the generalized Bessel function J−μn that is a regular integral of the first kind of the generalized Bessel differential equation (1) must be replaced by a generalized Neumann function Ynμ that is a regular integral of the second kind (Section 3.1) and is linearly independent, leading by and Y]μ(z) to the general integral linear combination (Section 3.2)
The generalized Bessel differential equation appears for coupled acoustic-vortical wave problems, which would have satisfied the original Bessel differential equation in the decoupled acoustic or vortical case
Summary
The Bessel differential equation was first considered in connexion with the oscillations of a heavy chain [1] and vibrations of a circular membrane [2] and has had since [3] a vast number of applications supported by an extensive theory [4]. The consideration of coupled acoustic-vortical waves as rotational compressible perturbations of a uniform mean flow with rigid body rotation leads to the generalized Bessel equation that differs from the original in having an extra term involving a second parameter, namely, the degree μ, in addition to the order ]. The solutions of the generalized Bessel differential equation around the regular singularity at the origin has (i) indices that are exponents of the leading power depending only on the order; (ii) recurrence relation for the coefficients of the power series. From (ii) it follows that the series expansion for the generalized Bessel (Section 2.1) and Neumann (Section 3.1) functions differ from the original series in having finite products multiplying each term; these finite products can be expressed as ratios of Gamma functions, whose arguments become singular for zero degree. The Wronskians are used to select pairs of linearly independent particular integrals that lead to the general integral for noninteger (Section 2.2) and integer (Section 3.2) order
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