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Recurrence Legendre Polynomials
Moscow University Mathematics Bulletin, 2022zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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Archiv der Mathematik, 2022
Let \(\mathcal{P}\) be the vector space of all polynomials, equipped with the inner product \(\langle f(x), g(x)\rangle=\int_{-1}^{1} f(x) g(x) d x\). The Legendre polynomials \(P_{0}(x), P_{1}(x), \ldots\) are the polynomials obtained by applying the Gram-Schmidt procedure to the ordered basis \(\mathcal{B}=\left\{1, x, x^{2}, \ldots\right\}\) of ...
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Let \(\mathcal{P}\) be the vector space of all polynomials, equipped with the inner product \(\langle f(x), g(x)\rangle=\int_{-1}^{1} f(x) g(x) d x\). The Legendre polynomials \(P_{0}(x), P_{1}(x), \ldots\) are the polynomials obtained by applying the Gram-Schmidt procedure to the ordered basis \(\mathcal{B}=\left\{1, x, x^{2}, \ldots\right\}\) of ...
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Legendre Polynomials and Legendre Functions
2021Legendre polynomials and, respectively, Legendre functions are one of the most important functions in physics. In this chapter, we will discuss and derive corresponding program codes supporting complex arguments and complex indices. The code is freely available.
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An inequality for Legendre polynomials
Journal of Mathematical Physics, 1994The following inequality is established: ‖Pn(cos ϑ)‖< [√1+(π4/16)(n+1/2)4 sin4 ϑ]−1, 0<ϑ<π, n=1,2,..., where Pn(x) denotes the Legendre polynomial of degree n. The relation P2n(cos ϑ) + (4/π2)× Q2n(cos ϑ) < [√1+(π4/16)(n+1/2)4 sin4 ϑ]−1, n=1,2,..., on [θn1,θn,n+1], is proven where Qn(x) denotes the Legendre function of ...
LAFORGIA, Andrea Ivo Antonio, Elbert, A.
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A Note on Legendre Polynomials
International Journal of Nonlinear Sciences and Numerical Simulation, 2001Summary: We use an operational method to show that Legendre polynomials can be viewed as discrete convolutions of Laguerre polynomials. It is furthermore shown that they can be derived as the particular case of a new family of two-variable orthogonal polynomials, whose properties are studied with some detail.
Dattoli, Giuseppe +2 more
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Associated Legendre Polynomial Approximations
Journal of Applied Physics, 1951Approximations for the associated Legendre Polynomials are derived by a phase integral method. The method is an extension of the WBK method, applicable to separable multidimensional wave propagation problems.
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1976
Publisher Summary This chapter focuses on Legendre's polynomials. It discusses Kodaira's identity, Weyl's theory, Green's formula, symmetric boundary conditions, T-positive theory, S-positive theory, and other theorems.
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Publisher Summary This chapter focuses on Legendre's polynomials. It discusses Kodaira's identity, Weyl's theory, Green's formula, symmetric boundary conditions, T-positive theory, S-positive theory, and other theorems.
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Expanding Legendre polynomials
Journal of Applied AnalysisAbstract The expansion of Legendre polynomials P ℓ
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From local explanations to global understanding with explainable AI for trees
Nature Machine Intelligence, 2020Scott M Lundberg +2 more
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