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The Singularity of Legendre Functions of the First Kind as a Consequence of the Symmetry of Legendre’s Equation
Symmetry ( IF 2.2 ) Pub Date : 2022-04-04 , DOI: 10.3390/sym14040741
Ramses van der Toorn

Legendre’s equation is key in various branches of physics. Its general solution is a linear function space, spanned by the Legendre functions of the first and second kind. In physics, however, commonly the only acceptable members of this set are Legendre polynomials. The quantization of the eigenvalues of Legendre’s operator is a consequence of this. We present and explain a stand-alone and in-depth argument for rejecting all solutions of Legendre’s equation in physics apart from the polynomial ones. We show that the combination of the linearity, the mirror symmetry and the signature of the regular singular points of Legendre’s equation are quintessential to the argument. We demonstrate that the evenness or oddness of Legendre polynomials is a consequence of the same premises.

中文翻译:

第一类勒让德函数的奇点是勒让德方程对称性的结果

勒让德方程是物理学各个分支的关键。它的通解是一个线性函数空间,由第一类和第二类勒让德函数跨越。然而,在物理学中,这个集合中唯一可接受的成员通常是勒让德多项式。勒让德算子的特征值的量化就是由此产生的结果。我们提出并解释了一个独立且深入的论点,以拒绝除了多项式之外的勒让德方程在物理学中的所有解。我们证明了勒让德方程的线性、镜像对称和正则奇异点的特征的组合是该论点的典型。我们证明勒让德多项式的偶数或奇数是相同前提的结果。
更新日期:2022-04-04
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