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Hace 2 días · Legendre's equation comes up in many physical situations involving spherical symmetry. Legendre Polynomials. Legendre's polynomial can be defined explicitly: Pn(x) = 1 2n ∑k=0⌊n/2⌋ (−1)k(n k)(2n − 2k n)xn−2k, (2) (2) P n ( x) = 1 2 n ∑ k = 0 ⌊ n / 2 ⌋ ( − 1) k ( n k) ( 2 n − 2 k n) x n − 2 k,
Hace 5 días · The Legendre symbol of \(a\) with respect to \(p\) is defined by \[\left(\dfrac{a}{p}\right)=\begin{cases}1 & \text{ if } a \text{ is a quadratic residue modulo } p \text{ and } a \not\equiv 0\pmod{p}\\-1 & \text{ if } a \text{ is a quadratic non-residue modulo } p\\0 & \text{ if } a \equiv 0 \pmod{p}.\end{cases}\]
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Hace 4 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
Hace 2 días · These angular solutions : are a product of trigonometric functions, here represented as a complex exponential, and associated Legendre polynomials: Y ℓ m ( θ , φ ) = N e i m φ P ℓ m ( cos θ ) {\displaystyle Y_{\ell }^{m}(\theta ,\varphi )=Ne^{im\varphi }P_{\ell }^{m}(\cos {\theta })}
15 de may. de 2024 · Legendre's polynomials are orthogonal. ∫1 −1Pn(x)Pm(x)dx ={0, 22n+1, for n ≠ m, for n = m. (2) (2) ∫ − 1 1 P n ( x) P m ( x) d x = { 0, for n ≠ m, 2 2 n + 1, for n = m. ∫π 0 Pmn (cos θ)Pm k (cos θ) sin θdθ = 2(k + m)! (2k + 1)(k − m)! δn,k ∫ 0 π P n m ( cos. θ) P k m ( cos. θ) sin. θ d θ = 2 ( k + m)! ( 2 k + 1) ( k − m)! δ n, k.
16 de may. de 2024 · See. Spherical Harmonic Addition Theorem. About MathWorld; MathWorld Classroom; Contribute; MathWorld Book; wolfram.com; 13,150 Entries
