On the Regular Integral Solutions of a Generalized Bessel Differential Equation

• Main Authors: L. M. B. C. Campos, F. Moleiro, M. J. S. Silva, J. Paquim
• Format: Article
• Language: English
• Published: Hindawi Limited 2018-01-01
• Series: Advances in Mathematical Physics
• Online Access: http://dx.doi.org/10.1155/2018/8919516

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.