2022/10/20 by F. M. S. Lima, Lima, F. M. S.
Mathematics · Physics and Astronomy · #42C10 #Algebraic and Geometric Analysis #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Number Theory (math.NT) #Relativity and Gravitational Theory
paper · pdf · doi:10.48550/arxiv.2210.10942
openalex publication_date 2022/10/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
As is well-known, the separation of variables in second order partial differential equations (PDEs) for physical problems with spherical symmetry usually leads to Cauchy's differential equation for the radial coordinate r and Legendre's differential equation for the polar angle θ. For eigenvalues of the form n (n+1), n ≥ 0 being an integer, Legendre's equation admits certain polynomials Pn(cosθ) as solutions, which form a complete set of continuous orthogonal functions for all θ∈ [0,π]. This allows us to take the polynomials Pn(x), where x = cosθ, as a basis for the Fourier-Legendre series expansion of any function f(x) continuous by parts over x ∈ [-1,1]. These lecture notes correspond to the end of my course on Mathematical Methods for Physics, when I did derive the differential equations and solutions for physical problems with spherical symmetry. For those interested in Number Theory, I have included an application of shifted Legendre polynomials in irrationality proofs, following a method introduced by Beukers to show that ζ(2) and ζ(3) are irrational numbers.