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In mathematics, Legendre functions are solutions to Legendre's differential equation: They are named after Adrien-Marie Legendre. This 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. The Legendre differential equation may be solved using the standard power series method. The equation has regular singular points at ''x'' = ±1 so, in general, a series solution about the origin will only converge for |''x''| < 1. When ''n'' is an integer, the solution ''P''''n''(''x'') that is regular at ''x'' = 1 is also regular at ''x'' = −1, and the series for this solution terminates (i.e. it is a polynomial). These solutions for ''n'' = 0, 1, 2, ... (with the normalization ''Pn''(1) = 1) form a polynomial sequence of orthogonal polynomials called the Legendre polynomials. Each Legendre polynomial ''P''''n''(''x'') is an ''n''th-degree polynomial. It may be expressed using Rodrigues' formula: : That these polynomials satisfy the Legendre differential equation () follows by differentiating ''n'' + 1 times both sides of the identity : and employing the general Leibniz rule for repeated differentiation. The ''P''''n'' can also be defined as the coefficients in a Taylor series expansion: = \sum_^\infty P_n(x) t^n.|}} In physics, this ordinary generating function is the basis for multipole expansions. == Recursive definition == Expanding the Taylor series in Equation () for the first two terms gives : for the first two Legendre Polynomials. To obtain further terms without resorting to direct expansion of the Taylor series, equation (2) is differentiated with respect to t on both sides and rearranged to obtain : Replacing the quotient of the square root with its definition in (), and equating the coefficients of powers of t in the resulting expansion gives ''Bonnet’s recursion formula'' : This relation, along with the first two polynomials ''P''0 and ''P''1, allows the Legendre Polynomials to be generated recursively. Explicit representations include : where the latter, which is immediate from the recursion formula, expresses the Legendre polynomials by simple monomials and involves the multiplicative formula of the binomial coefficient. The first few Legendre polynomials are: : The graphs of these polynomials (up to ''n'' = 5) are shown below: 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Legendre polynomials」の詳細全文を読む スポンサード リンク
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