Topic
Legendre polynomials
About: Legendre polynomials is a research topic. Over the lifetime, 6133 publications have been published within this topic receiving 100536 citations. The topic is also known as: Legendre polynomials.
Papers published on a yearly basis
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Papers
Book•
1 Jan 1947
TL;DR: In this paper, the authors present an algebraic extension of LINEAR TRANSFORMATIONS and QUADRATIC FORMS, and apply it to EIGEN-VARIATIONS.
Abstract: Partial table of contents: THE ALGEBRA OF LINEAR TRANSFORMATIONS AND QUADRATIC FORMS. Transformation to Principal Axes of Quadratic and Hermitian Forms. Minimum-Maximum Property of Eigenvalues. SERIES EXPANSION OF ARBITRARY FUNCTIONS. Orthogonal Systems of Functions. Measure of Independence and Dimension Number. Fourier Series. Legendre Polynomials. LINEAR INTEGRAL EQUATIONS. The Expansion Theorem and Its Applications. Neumann Series and the Reciprocal Kernel. The Fredholm Formulas. THE CALCULUS OF VARIATIONS. Direct Solutions. The Euler Equations. VIBRATION AND EIGENVALUE PROBLEMS. Systems of a Finite Number of Degrees of Freedom. The Vibrating String. The Vibrating Membrane. Green's Function (Influence Function) and Reduction of Differential Equations to Integral Equations. APPLICATION OF THE CALCULUS OF VARIATIONS TO EIGENVALUE PROBLEMS. Completeness and Expansion Theorems. Nodes of Eigenfunctions. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS. Bessel Functions. Asymptotic Expansions. Additional Bibliography. Index.
8,457 citations
1 Jan 1977
TL;DR: Spectral Methods Survey of Approximation Theory Review of Convergence Theory Algebraic Stability Spectral Methods Using Fourier Series Applications of algebraic stability analysis Constant Coefficient Hyperbolic Equations Time Differencing Efficient Implementation of Spectral Method as discussed by the authors.
Abstract: Spectral Methods Survey of Approximation Theory Review of Convergence Theory Algebraic Stability Spectral Methods Using Fourier Series Applications of Algebraic Stability Analysis Constant Coefficient Hyperbolic Equations Time Differencing Efficient Implementation of Spectral Methods Numerical Results for Hyperbolic Problems Advection-Diffusion Equation Models of Incompressible Fluid Dynamics Miscellaneous Applications of Spectral Methods Survey of Spectral Methods and Applications Properties of Chebyshev and Legendre Polynomial Expansions.
3,797 citations
TL;DR: In this paper, an approach to numerical convection is presented that exclusively yields upstream-centered schemes, which start from a meshwise approximation of the initial-value distribution by simple basic functions, e.g., Legendre polynomials.
2,495 citations
Book•
26 Sep 1986
TL;DR: This chapter discusses the development of Mathematical Statistics in Astronomy and Geodesy before 1827 and some of the ideas behind Laplace's Rescue of the Solar System.
Abstract: Introduction PART 1: The Development of Mathematical Statistics in Astronomy and Geodesy before 1827 1. Least Squares and the Combination of Observations Legendre in 1805 Cotes's Rule Tobias Mayer and the Libration of the Moon Saturn, Jupiter, and Enter Laplace's Rescue of the Solar System Roger Boscovich and the Figure of the Earth Laplace and the Method of Situation Legendre and the Invention of Least Squares 2. Probabilists and the Measurement of Uncertainty Jacob Bernoulli De Moivre and the Expanded Binomial Bernoulli's Failure De Moivre's Approximation De Moivre's Deficiency Simpson and Bayes Simpson's Crucial Step toward Error A Bayesian Critique 3. Inverse Probability Laplace and Inverse Probability The Choice of Means The Deduction of a Curve of Errors in 1772-1774
1,072 citations
TL;DR: In this paper, the angular distribution in collisions between pairs of particles is simplified by performing explicitly all sums over magnetic quantum numbers, and the resulting expressions involve coefficients introduced by Racah for the study of complex atomic spectra.
Abstract: The general formula for the angular distribution in collisions between pairs of particles is simplified by performing explicitly all sums over magnetic quantum numbers. The resulting expressions involve coefficients introduced by Racah for the study of complex atomic spectra. The cross sections are expressed as series in Legendre polynomials, each coefficient in the series being manifestly real.The general theory is then specialized for the case of nuclear reactions and scattering associated with one isolated resonance level of the compound nucleus. Formulas are derived for the various differential reaction cross sections and for scattering with and without change of channel spin. The interference terms between resonance and potential scattering are written explicitly, both for neutral and for charged particles.
783 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2026 | 4 |
| 2025 | 103 |
| 2024 | 226 |
| 2023 | 306 |
| 2022 | 652 |
| 2021 | 322 |