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Home > Legendre polynomials

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 solution is finite (i.e. the series converges) provided |x| < 1. Furthermore, it is finite at x = ± 1 provided n is a non-negative integer, i.e. n = 0, 1, 2,... . In this case, the solutions form a polynomial sequence called the Legendre polynomials.

Each Legendre polynomial P_n(x) is an nth-degree polynomial. It may be expressed using Rodrigues' formula:

An important property of the Legendre polynomials is that they are orthogonal with respect to the L² inner product on the interval −1 ≤ x ≤ 1:

(where δ_mn denotes the Kronecker delta, equal to 1 if m = n and to 0 otherwise). In fact, an alternative derivation of the Legendre polynomials is by carrying out the Gram-Schmidt processIn mathematics and numerical analysis, the Gram-Schmidt process of linear algebra is a method of orthogonalizing a set of vectors in an inner product space, most commonly the Euclidean space R n''. Orthogonalization in this context means the following: we on the polynomials {1, x, x², ...} with respect to this inner product.

These are the first few Legendre polynomials:

The graphs of these polynomials (up to n=5) are shown below:

References

• M. Abramowitz and I. A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. (See Chapter 8.)

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Legendre symbolThe Legendre symbol is used by mathematicians in the area of number theory, particularly in the fields of factorization and quadratic residues. It is named after the French mathematician Adrien-Marie Legendre. Definition The Legendre symbol is a special c Legendre polynomialsNote: The term Legendre polynomials is sometimes used (wrongly) to indicate the associated Legendre polynomials. In mathematics, Legendre functions are solutions to Legendre's differential equation : They are named after Adrien-Marie Legendre. This ordina Legendre transformationIn mathematics, two differentiable functions f and g are said to be Legendre transforms of each other if their first derivatives are inverse functions of each other: : f and g are then said to be related by a Legendre transformation . Legendre transformat Legendre formIn mathematics, the Legendre forms of elliptic integrals F φ k , E φ k and P φ k ''n are defined by : : :. Legendre's constantLegendre's constant is a "phantom" that doesn't really exist. Before the discovery of the prime number theorem, examination of available numerical evidence for known primes had led Legendre to conjecture that perhaps : where : The quantity 1. was called L Legendre (crater)Legendre is a lunar impact crater that is located near the eastern limb of the Moon. Just to the southwest is the Adams crater. To the northwest is Palitzsch and the prominent Petavius crater. The rim of Legendre is worn and eroded, with many small crater