The orthogonality and completeness of these solutions is best seen from the viewpoint of Sturm—Liouville theory. We rewrite the differential equation as an eigenvalue problem,.
The orthogonality and completeness of this set of solutions follows at once from the larger framework of Sturm—Liouville theory. A two-parameter generalization of Eq. Legendre functions are solutions of Legendre's differential equation generalized or not with non-integer parameters. In physical settings, Legendre's differential equation arises naturally whenever one solves Laplace's equation and related partial differential equations by separation of variables in spherical coordinates.
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From this standpoint, the eigenfunctions of the angular part of the Laplacian operator are the spherical harmonics , of which the Legendre polynomials are up to a multiplicative constant the subset that is left invariant by rotations about the polar axis. This approach to the Legendre polynomials provides a deep connection to rotational symmetry.
Many of their properties which are found laboriously through the methods of analysis — for example the addition theorem — are more easily found using the methods of symmetry and group theory, and acquire profound physical and geometrical meaning. Since they are also orthogonal with respect to the same norm, the two statements can be combined into the single equation,. This normalization is most readily found by employing Rodrigues' formula , given below.
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That the polynomials are complete means the following. This completeness property underlies all the expansions discussed in this article, and is often stated in the form. An especially compact expression for the Legendre polynomials is given by Rodrigues' formula :. Among these are explicit representations such as.
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The Legendre polynomials were first introduced in by Adrien-Marie Legendre  as the coefficients in the expansion of the Newtonian potential. The expression gives the gravitational potential associated to a point mass or the Coulomb potential associated to a point charge.
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The expansion using Legendre polynomials might be useful, for instance, when integrating this expression over a continuous mass or charge distribution. A l and B l are to be determined according to the boundary condition of each problem. Legendre polynomials are also useful in expanding functions of the form this is the same as before, written a little differently :.
The left-hand side of the equation is the generating function for the Legendre polynomials. If the radius r of the observation point P is greater than a , the potential may be expanded in the Legendre polynomials. This expansion is used to develop the normal multipole expansion. Conversely, if the radius r of the observation point P is smaller than a , the potential may still be expanded in the Legendre polynomials as above, but with a and r exchanged. This expansion is the basis of interior multipole expansion. The first several orders are as follows:.
Generalized associated legendre functions and their applications
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Application of the new special functions allows one to increase considerably the number of problems whose solutions are found in a closed form, to examine these solutions, and to investigate the relationships between different classes of the special functions. This book deals with the theory and applications of generalized associated Legendre functions of the first and the second kind, P m, n? They occur as generalizations of classical Legendre functions of the first and the second kind respectively.