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Hace 6 días · The Legendre symbol is a function that encodes the information about whether a number is a quadratic residue modulo an odd prime. It is used in the law of quadratic reciprocity to simplify notation.
Hace 3 días · The corresponding Fourier--Legendre coefficients can be evaluated based on the formulas: \[ \int_{-1}^1 P_n (x)\,{\text d}x = 0 \qquad \mbox{and} \qquad \int_{-1}^a P_n (x)\,{\text d}x = \frac{1}{2n+1} \left[ P_{n+1} (a) - P_{n-1} (a) \right] .
16 de may. de 2024 · Legendre Identity -- from Wolfram MathWorld. Calculus and Analysis. Special Functions. Elliptic Integrals.
Hace 5 días · Legendre lets a and A represent positive primes ≡ 1 (mod 4) and b and B positive primes ≡ 3 (mod 4), and sets out a table of eight theorems that together are equivalent to quadratic reciprocity: Theorem
16 de may. de 2024 · See. Spherical Harmonic Addition Theorem. About MathWorld; MathWorld Classroom; Contribute; MathWorld Book; wolfram.com; 13,150 Entries
15 de may. de 2024 · Legendre's polynomials are orthogonal \begin{equation} \label{Eqlegendre.4} \int_{-1}^1 P_n (x) \, P_m (x) \,{\text d} x = \begin{cases} 0 , & \ \mbox{for} \quad n\ne m, \\ \frac{2}{2n+1} , & \ \mbox{for} \quad n = m. \end{cases} \end{equation}
Hace 3 días · This exact rule is known as the Gauss–Legendre quadrature rule. The quadrature rule will only be an accurate approximation to the integral above if f (x) is well-approximated by a polynomial of degree 2n − 1 or less on [−1, 1]. The Gauss–Legendre quadrature rule is not typically used for integrable functions with endpoint ...
