The original Bessel differential equation that describes, among many others, cylindrical acoustic or vortical waves, is a particular\ncase of zero degree of the generalized Bessel differential equation that describes coupled acoustic-vortical waves.The solutions of\nthe generalized Bessel differential equation are obtained for all possible combinations of the two complex parameters, order and\ndegree, and finite complex variable, as Frobenius-Fuchs series around the regular singularity at the origin; the series converge in the\nwhole complex plane of the variable, except for the point-at-infinity, that is, the only other singularity of the differential equation.\nThe regular integral solutions of the first and second kinds lead, respectively, to the generalized Bessel and Neumann functions;\nthese reduce to the original Bessel and Neumann functions for zero degree and have alternative expressions for nonzero degree
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