In physical science and
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Legendre polynomials (named after
Adrien-Marie Legendre
Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are named ...
, who discovered them in 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 function
In physical science and mathematics, the Legendre functions , and associated Legendre functions , , and Legendre functions of the second kind, , are all solutions of Legendre's differential equation. The Legendre polynomials and the associated L ...
s, Legendre functions of the second kind, and
associated Legendre functions.
Definition by construction as an orthogonal system
In this approach, the polynomials are defined as an orthogonal system with respect to the weight function
over the interval