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, 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 functions, Legendre functions of the second kind, and
associated Legendre function
In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation
\left(1 - x^2\right) \frac P_\ell^m(x) - 2 x \frac P_\ell^m(x) + \left \ell (\ell + 1) - \frac \rightP_\ell^m(x) = 0,
or equivalentl ...
s.
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