Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

The Singularity of Legendre Functions of the First Kind as a Consequence of the Symmetry of Legendre’s Equation

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)

A peer-reviewed article of this Preprint also exists.

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, doi:10.3390/sym14040741. 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, doi:10.3390/sym14040741.

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.

Keywords

Legendre’s Equation; Legendre Functions; Legendre Polynomials; Singularities; Symmetry

Subject

Physical Sciences, Mathematical Physics

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