[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/associated-legendre-polynomials-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki24\/associated-legendre-polynomials-wikipedia\/","headline":"Associated Legendre polynomials – Wikipedia","name":"Associated Legendre polynomials – Wikipedia","description":"Canonical solutions of the general Legendre equation In mathematics, the associated Legendre polynomials are the canonical solutions of the general","datePublished":"2020-06-09","dateModified":"2020-06-09","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/914d8bc4e6fd78db7db32609c345663927ad11ab","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/914d8bc4e6fd78db7db32609c345663927ad11ab","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki24\/associated-legendre-polynomials-wikipedia\/","about":["Wiki"],"wordCount":29859,"articleBody":"Canonical solutions of the general Legendre equationIn mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation(1\u2212x2)d2dx2P\u2113m(x)\u22122xddxP\u2113m(x)+[\u2113(\u2113+1)\u2212m21\u2212x2]P\u2113m(x)=0,{displaystyle left(1-x^{2}right){frac {d^{2}}{dx^{2}}}P_{ell }^{m}(x)-2x{frac {d}{dx}}P_{ell }^{m}(x)+left[ell (ell +1)-{frac {m^{2}}{1-x^{2}}}right]P_{ell }^{m}(x)=0,}or equivalentlyddx[(1\u2212x2)ddxP\u2113m(x)]+[\u2113(\u2113+1)\u2212m21\u2212x2]P\u2113m(x)=0,{displaystyle {frac {d}{dx}}left[left(1-x^{2}right){frac {d}{dx}}P_{ell }^{m}(x)right]+left[ell (ell +1)-{frac {m^{2}}{1-x^{2}}}right]P_{ell }^{m}(x)=0,}where the indices \u2113 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 [\u22121, 1] only if \u2113 and m are integers with 0 \u2264 m \u2264 \u2113, or with trivially equivalent negative values. When in addition m is even, the function is a polynomial. When m is zero and \u2113 integer, these functions are identical to the Legendre polynomials.  In general, when \u2113 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 \u2113 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.Table of ContentsDefinition for non-negative integer parameters \u2113 and m[edit]Alternative notations[edit]Closed Form[edit]Orthogonality[edit]Negative m and\/or negative \u2113[edit]The first few associated Legendre functions[edit]Recurrence formula[edit]Gaunt’s formula[edit]Generalization via hypergeometric functions[edit]Reparameterization in terms of angles[edit]Applications in physics: spherical harmonics[edit]Generalizations[edit]See also[edit]Notes and references[edit]External links[edit]Definition for non-negative integer parameters \u2113 and m[edit]These functions are denoted P\u2113m(x){displaystyle P_{ell }^{m}(x)}, where the superscript indicates the order and not a power of P.  Their most straightforward definition is in termsof derivatives of ordinary Legendre polynomials (m \u2265 0)P\u2113m(x)=(\u22121)m(1\u2212x2)m\/2dmdxm(P\u2113(x)),{displaystyle P_{ell }^{m}(x)=(-1)^{m}(1-x^{2})^{m\/2}{frac {d^{m}}{dx^{m}}}left(P_{ell }(x)right),}The (\u22121)m factor in this formula is known as the Condon\u2013Shortley 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 \u2113 and m follows by differentiating m times the Legendre equation for P\u2113:[1](1\u2212x2)d2dx2P\u2113(x)\u22122xddxP\u2113(x)+\u2113(\u2113+1)P\u2113(x)=0.{displaystyle left(1-x^{2}right){frac {d^{2}}{dx^{2}}}P_{ell }(x)-2x{frac {d}{dx}}P_{ell }(x)+ell (ell +1)P_{ell }(x)=0.}Moreover, since by Rodrigues’ formula,P\u2113(x)=12\u2113\u2113!\u00a0d\u2113dx\u2113[(x2\u22121)\u2113],{displaystyle P_{ell }(x)={frac {1}{2^{ell },ell !}} {frac {d^{ell }}{dx^{ell }}}left[(x^{2}-1)^{ell }right],}the Pm\u2113 can be expressed in the formP\u2113m(x)=(\u22121)m2\u2113\u2113!(1\u2212x2)m\/2\u00a0d\u2113+mdx\u2113+m(x2\u22121)\u2113.{displaystyle P_{ell }^{m}(x)={frac {(-1)^{m}}{2^{ell }ell !}}(1-x^{2})^{m\/2} {frac {d^{ell +m}}{dx^{ell +m}}}(x^{2}-1)^{ell }.}This equation allows extension of the range of m to: \u2212\u2113 \u2264 m \u2264 \u2113. The definitions of P\u2113\u00b1m, resulting from this expression by substitution of \u00b1m, are proportional. Indeed, equate the coefficients of equal powers on the left and right hand side ofd\u2113\u2212mdx\u2113\u2212m(x2\u22121)\u2113=clm(1\u2212x2)md\u2113+mdx\u2113+m(x2\u22121)\u2113,{displaystyle {frac {d^{ell -m}}{dx^{ell -m}}}(x^{2}-1)^{ell }=c_{lm}(1-x^{2})^{m}{frac {d^{ell +m}}{dx^{ell +m}}}(x^{2}-1)^{ell },}then it follows that the proportionality constant isclm=(\u22121)m(\u2113\u2212m)!(\u2113+m)!,{displaystyle c_{lm}=(-1)^{m}{frac {(ell -m)!}{(ell +m)!}},}so thatP\u2113\u2212m(x)=(\u22121)m(\u2113\u2212m)!(\u2113+m)!P\u2113m(x).{displaystyle P_{ell }^{-m}(x)=(-1)^{m}{frac {(ell -m)!}{(ell +m)!}}P_{ell }^{m}(x).}Alternative notations[edit]The following alternative notations are also used in literature:[2]P\u2113m(x)=(\u22121)mP\u2113m(x){displaystyle P_{ell m}(x)=(-1)^{m}P_{ell }^{m}(x)}Closed Form[edit]The Associated Legendre Polynomial can also be written as:Plm(x)=(\u22121)m\u22c52l\u22c5(1\u2212x2)m\/2\u22c5\u2211k=mlk!(k\u2212m)!\u22c5xk\u2212m\u22c5(lk)(l+k\u221212l){displaystyle P_{l}^{m}(x)=(-1)^{m}cdot 2^{l}cdot (1-x^{2})^{m\/2}cdot sum _{k=m}^{l}{frac {k!}{(k-m)!}}cdot x^{k-m}cdot {binom {l}{k}}{binom {frac {l+k-1}{2}}{l}}}with simple monomials and the generalized form of the binomial coefficient.Orthogonality[edit]The associated Legendre polynomials are not mutually orthogonal in general. For example, P11{displaystyle P_{1}^{1}} is not orthogonal to P22{displaystyle P_{2}^{2}}. However, some subsets are orthogonal. Assuming 0\u00a0\u2264\u00a0m\u00a0\u2264\u00a0\u2113, they satisfy the orthogonality condition for fixed m:\u222b\u221211PkmP\u2113mdx=2(\u2113+m)!(2\u2113+1)(\u2113\u2212m)!\u00a0\u03b4k,\u2113{displaystyle int _{-1}^{1}P_{k}^{m}P_{ell }^{m}dx={frac {2(ell +m)!}{(2ell +1)(ell -m)!}} delta _{k,ell }}Where \u03b4k,\u2113 is the Kronecker delta.Also, they satisfy the orthogonality condition for fixed \u2113:\u222b\u221211P\u2113mP\u2113n1\u2212x2dx={0if\u00a0m\u2260n(\u2113+m)!m(\u2113\u2212m)!if\u00a0m=n\u22600\u221eif\u00a0m=n=0{displaystyle int _{-1}^{1}{frac {P_{ell }^{m}P_{ell }^{n}}{1-x^{2}}}dx={begin{cases}0&{text{if }}mneq n\\{frac {(ell +m)!}{m(ell -m)!}}&{text{if }}m=nneq 0\\infty &{text{if }}m=n=0end{cases}}}Negative m and\/or negative \u2113[edit]The differential equation is clearly invariant under a change in sign of m.The functions for negative m were shown above to be proportional to those of positive m:P\u2113\u2212m=(\u22121)m(\u2113\u2212m)!(\u2113+m)!P\u2113m{displaystyle P_{ell }^{-m}=(-1)^{m}{frac {(ell -m)!}{(ell +m)!}}P_{ell }^{m}}(This followed from the Rodrigues’ formula definition.  This definition also makes the various recurrence formulas work for positive or negative m.)"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/associated-legendre-polynomials-wikipedia\/#breadcrumbitem","name":"Associated Legendre polynomials – Wikipedia"}}]}]