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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.


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Book
01 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.

7,426 citations

MonographDOI
01 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.

3,386 citations

Journal ArticleDOI
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.

2,206 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

1,066 citations

Book
01 Jan 1966
TL;DR: Inverse Trigonometric and Hyperbolic Functions as mentioned in this paper, the exponential and trigonometric functions of complex numbers are used to define the series of positive terms in the complex number space.
Abstract: Chapter 1: Infinite Series, Power Series.The Geometric Series.Definitions and Notation.Applications of Series.Convergent and Divergent Series.Convergence Tests.Convergence Tests for Series of Positive Terms.Alternating Series.Conditionally Convergent Series.Useful Facts about Series.Power Series Interval of Convergence.Theorems about Power Series.Expanding Functions in Power Series.Expansion Techniques.Accuracy of Series Approximations.Some Uses of Series.Chapter 2: Complex Numbers.Introduction.Real and Imaginary Parts of a Complex Number.The Complex Plane.Terminology and Notation.Complex Algebra.Complex Infinite Series.Complex Power Series Disk of Convergence.Elementary Functions of Complex Numbers.Euler's Formula.Powers and Roots of Complex Numbers.The Exponential and Trigonometric Functions.Hyperbolic Functions.Logarithms.Complex Roots and Powers.Inverse Trigonometric and Hyperbolic Functions.Some Applications.Chapter 3: Linear Algebra.Introduction.Matrices Row Reduction.Determinants Cramer's Rule.Vectors.Lines and Planes.Matrix Operations.Linear Combinations, Functions, Operators.Linear Dependence and Independence.Special Matrices and Formulas.Linear Vector Spaces.Eigenvalues and Eigenvectors.Applications of Diagonalization.A Brief Introduction to Groups.General Vector Spaces.Chapter 4: Partial Differentiation.Introduction and Notation.Power Series in Two Variables.Total Differentials.Approximations using Differentials.Chain Rule.Implicit Differentiation.More Chain Rule.Maximum and Minimum Problems.Constraints Lagrange Multipliers.Endpoint or Boundary Point Problems.Change of Variables.Differentiation of Integrals.Chapter 5: Multiple Integrals.Introduction.Double and Triple Integrals.Applications of Integration.Change of Variables in Integrals Jacobians.Surface Integrals.Chapter 6: Vector Analysis.Introduction.Applications of Vector Multiplication.Triple Products.Differentiation of Vectors.Fields.Directional Derivative Gradient.Some Other Expressions Involving V.Line Integrals.Green's Theorems in the Plane.The Divergence and the Divergence Theorem.The Curl and Stokes' Theorem.Chapter 7: Fourier Series and Transforms.Introduction.Simple Harmonic Motion and Wave Motion Periodic Functions.Applications of Fourier Series.Average Value of a Function.Fourier Coefficients.Complex Form of Fourier Series.Other Intervals.Even and Odd Functions.An Application to Sound.Parseval's Theorem.Fourier Transforms.Chapter 8: Ordinary Differential Equations.Introduction.Separable Equations.Linear First-Order Equations.Other Methods for First-Order Equations.Linear Equations (Zero Right-Hand Side).Linear Equations (Nonzero Right-Hand Side).Other Second-Order Equations.The Laplace Transform.Laplace Transform Solutions.Convolution.The Dirac Delta Function.A Brief Introduction to Green's Functions.Chapter 9: Calculus of Variations.Introduction.The Euler Equation.Using the Euler Equation.The Brachistochrone Problem Cycloids.Several Dependent Variables Lagrange's Equations.Isoperimetric Problems.Variational Notation.Chapter 10: Tensor Analysis.Introduction.Cartesian Tensors.Tensor Notation and Operations.Inertia Tensor.Kronecker Delta and Levi-Civita Symbol.Pseudovectors and Pseudotensors.More about Applications.Curvilinear Coordinates.Vector Operators.Non-Cartesian Tensors.Chapter 11: Special Functions.Introduction.The Factorial Function.Gamma Function Recursion Relation.The Gamma Function of Negative Numbers.Formulas Involving Gamma Functions.Beta Functions.Beta Functions in Terms of Gamma Functions.The Simple Pendulum.The Error Function.Asymptotic Series.Stirling's Formula.Elliptic Integrals and Functions.Chapter 12: Legendre, Bessel, Hermite, and Laguerre functions.Introduction.Legendre's Equation.Leibniz' Rule for Differentiating Products.Rodrigues' Formula.Generating Function for Legendre Polynomials.Complete Sets of Orthogonal Functions.Orthogonality of Legendre Polynomials.Normalization of Legendre Polynomials.Legendre Series.The Associated Legendre Polynomials.Generalized Power Series or the Method of Frobenius.Bessel's Equation.The Second Solutions of Bessel's Equation.Graphs and Zeros of Bessel Functions.Recursion Relations.Differential Equations with Bessel Function Solutions.Other Kinds of Bessel Functions.The Lengthening Pendulum.Orthogonality of Bessel Functions.Approximate Formulas of Bessel Functions.Series Solutions Fuch's Theorem.Hermite and Laguerre Functions Ladder Operators.Chapter 13: Partial Differential Equations.Introduction.Laplace's Equation Steady-State Temperature.The Diffusion of Heat Flow Equation the Schrodinger Equation.The Wave Equation the Vibrating String.Steady-State Temperature in a Cylinder.Vibration of a Circular Membrane.Steady-State Temperature in a Sphere.Poisson's Equation.Integral Transform Solutions of Partial Differential Equations.Chapter 14: Functions of a Complex Variable.Introduction.Analytic Functions.Contour Integrals.Laurent Series.The Residue Theorem.Methods of Finding Residues.Evaluation of Definite Integrals.The Point at Infinity Residues of Infinity.Mapping.Some Applications of Conformal Mapping.Chapter 15: Probability and Statistics.Introduction.Sample Space.Probability Theorems.Methods of Counting.Random Variables.Continuous Distributions.Binomial Distribution.The Normal or Gaussian Distribution.The Poisson Distribution.Statistics and Experimental Measurements.

692 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
2023212
2022493
2021313
2020295
2019276
2018294