Legendre polynomials

In physical science and 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 mathematically beautiful 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. 1 1 − 2 x t + t 2 = ∑ n = 0 ∞ P n ( x ) t n . {displaystyle {frac {1}{sqrt {1-2xt+t^{2}}}}=sum _{n=0}^{infty }P_{n}(x)t^{n},.}     (2) d d x [ ( 1 − x 2 ) d P n ( x ) d x ] + n ( n + 1 ) P n ( x ) = 0 . {displaystyle {frac {d}{dx}}left+n(n+1)P_{n}(x)=0,.}     (1) In physical science and 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 mathematically beautiful 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. In this approach, the polynomials are defined as an orthogonal system with respect to the weight function w ( x ) = 1 {displaystyle w(x)=1} over the interval [ − 1 , 1 ] {displaystyle } . That is, P n ( x ) {displaystyle P_{n}(x)} is a polynomial of degree n {displaystyle n} , such that This determines the polynomials completely up to an overall scale factor, which is fixed by the standardization P n ( 1 ) = 1 {displaystyle P_{n}(1)=1} . That this is a constructive definition is seen thus: P 0 ( x ) = 1 {displaystyle P_{0}(x)=1} is the only correctly standardized polynomial of degree 0. P 1 ( x ) {displaystyle P_{1}(x)} must be orthogonal to P 0 {displaystyle P_{0}} , leading to P 1 ( x ) = x {displaystyle P_{1}(x)=x} , and P 2 ( x ) {displaystyle P_{2}(x)} is determined by demanding orthogonality to P 0 {displaystyle P_{0}} and P 1 {displaystyle P_{1}} , and so on. P n {displaystyle P_{n}} is fixed by demanding orthogonality to all P m {displaystyle P_{m}} with m < n {displaystyle m<n} . This gives n {displaystyle n} conditions, which, along with the standardization P n ( 1 ) = 1 {displaystyle P_{n}(1)=1} fixes all n + 1 {displaystyle n+1} coefficients in P n ( x ) {displaystyle P_{n}(x)} . With work, all the coefficients of every polynomial can be systematically determined, leading to the explicit representation in powers of x {displaystyle x} given below. This definition of the P n {displaystyle P_{n}} 's is the simplest one. It does not appeal to the theory of differential equations. Second, the completeness of the polynomials follows immediately from the completeness of the powers 1, x , x 2 , x 3 , … {displaystyle x,x^{2},x^{3},ldots } . Finally, by defining them via orthogonality with respect to the most obvious weight function on a finite interval, it sets up the Legendre polynomials as one of the three classical orthogonal polynomial systems. The other two are the Laguerre polynomials, which are orthogonal over the half line [ 0 , ∞ ) {displaystyle [0,infty )} , and the Hermite polynomials, orthogonal over the full line ( − ∞ , ∞ ) {displaystyle (-infty ,infty )} , with weight functions that are the most natural analytic functions that ensure convergence of all integrals. The Legendre polynomials can also be defined as the coefficients in a formal expansion in powers of t {displaystyle t} of the generating function The coefficient of t n {displaystyle t^{n}} is a polynomial in x {displaystyle x} of degree n {displaystyle n} . Expanding up to t 1 {displaystyle t^{1}} gives