**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Concept# Associated Legendre polynomials

Summary

In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation
or equivalently
where the indices l and m (which are integers) are referred to as the degree and order of the associated Legendre polynomial respectively. This equation has nonzero solutions that are nonsingular on only if l and m are integers with 0 ≤ m ≤ l, or with trivially equivalent negative values. When in addition m is even, the function is a polynomial. When m is zero and l integer, these functions are identical to the Legendre polynomials. In general, when l and m are integers, the regular solutions are sometimes called "associated Legendre polynomials", even though they are not polynomials when m is odd. The fully general class of functions with arbitrary real or complex values of l and m are Legendre functions. In that case the parameters are usually labelled with Greek letters.
The Legendre ordinary differential equation is frequently encountered in physics and other technical fields. In particular, it occurs when solving Laplace's equation (and related partial differential equations) in spherical coordinates. Associated Legendre polynomials play a vital role in the definition of spherical harmonics.
These functions are denoted , where the superscript indicates the order and not a power of P. Their most straightforward definition is in terms
of derivatives of ordinary Legendre polynomials (m ≥ 0)
The (−1)m factor in this formula is known as the Condon–Shortley phase. Some authors omit it. That the functions described by this equation satisfy the general Legendre differential equation with the indicated values of the parameters l and m follows by differentiating m times the Legendre equation for Pl:
Moreover, since by Rodrigues' formula,
the P can be expressed in the form
This equation allows extension of the range of m to: −l ≤ m ≤ l. The definitions of Pl±m, resulting from this expression by substitution of ±m, are proportional.

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (15)

Related people (1)

Related units (2)

Related concepts (2)

Related courses (7)

Related lectures (33)

In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, each function defined on the surface of a sphere can be written as a sum of these spherical harmonics. This is similar to periodic functions defined on a circle that can be expressed as a sum of circular functions (sines and cosines) via Fourier series.

In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a vast number of mathematical properties and numerous applications. They can be defined in many ways, and the various definitions highlight different aspects as well as suggest generalizations and connections to different mathematical structures and physical and numerical applications. Closely related to the Legendre polynomials are associated Legendre polynomials, Legendre functions, Legendre functions of the second kind, and associated Legendre functions.

En son coeur, c'est un cours d'analyse fonctionnelle pour les physiciens et traite les bases de théorie de mesure, des espaces des fonctions et opérateurs linéaires.

Le but du cours de physique générale est de donner à l'étudiant les notions de base nécessaires à la compréhension des phénomènes physiques. L'objectif est atteint lorsque l'étudiant est capable de pr

The main goal of this course is to give the student a solid introduction into approaches, methods, and
instrumentation used in biomedical research. A major focus is on Magnetic Resonance Imaging (MRI)

Victor Panaretos, Julien René Pierre Fageot, Matthieu Martin Jean-André Simeoni, Alessia Caponera

We propose nonparametric estimators for the second-order central moments of possibly anisotropic spherical random fields, within a functional data analysis context. We consider a measurement framework where each random field among an identically distribute ...

2022Polynomial Approximation: Orthonormal Basis and ProjectionMOOC: Digital Signal Processing I

Explores polynomial approximation with orthonormal bases and orthogonal projection methods.

Constraints, Power, Work, and Kinetic EnergyPHYS-101(f): General physics : mechanics

Covers the general oscillation period equation, initial conditions, integration, elliptic integrals, Legendre polynomials, work, kinetic energy, and power.

Mathematical Pendulum

Explores the mathematical pendulum, its oscillations, and period calculation.

We consider the singular set in the thin obstacle problem with weight vertical bar x(n +1)vertical bar(a) for a epsilon (-1, 1), which arises as the local extension of the obstacle problem for the fractional Laplacian (a nonlocal problem). We develop a ref ...

We measure the anisotropic clustering of the quasar sample from Data Release 16 (DR16) of the Sloan Digital Sky Survey IV extended Baryon Oscillation Spectroscopic Survey (eBOSS). A sample of 343 708 spectroscopically confirmed quasars between redshift 0.8 ...