Version 1
: Received: 27 January 2022 / Approved: 28 January 2022 / Online: 28 January 2022 (13:23:54 CET)
Version 2
: Received: 6 March 2022 / Approved: 11 March 2022 / Online: 11 March 2022 (09:33:45 CET)
van der Toorn, R. The Singularity of Legendre Functions of the First Kind as a Consequence of the Symmetry of Legendre’s Equation. Symmetry2022, 14, 741.
van der Toorn, R. The Singularity of Legendre Functions of the First Kind as a Consequence of the Symmetry of Legendre’s Equation. Symmetry 2022, 14, 741.
van der Toorn, R. The Singularity of Legendre Functions of the First Kind as a Consequence of the Symmetry of Legendre’s Equation. Symmetry2022, 14, 741.
van der Toorn, R. The Singularity of Legendre Functions of the First Kind as a Consequence of the Symmetry of Legendre’s Equation. Symmetry 2022, 14, 741.
Abstract
Legendre’s equation is key in various branches of physics. Its general solution is a linear function space, spanned by the Legendre functions of first and second kind. In physics however, commonly the only acceptable members of this set are the Legendre polynomials. Quantization of the eigenvalues of Legendre’s operator is a consequence of this. We present and explain a stand-alone, in-depth argument for rejecting all solutions of Legendre’s equation, but the polynomial ones, in physics. We show that the combination of the linearity, the mirror symmetry and the signature of the regular singular points of Legendre’s equation is quintessential to the argument. We demonstrate that the evenness or oddness of the Legendre polynomials is a consequence of the same ingredients.
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
