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Hace 3 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 4 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 ⌊ n / 2 ⌋ ∑ k = 0 ( − 1)k(n k)(2n − 2k n)xn − 2k, where ⌊ · ⌋ is the floor function, and (n k) = n! k!(n − k)! = nk _ k! is the binomial coefficient.
Hace 4 días · Legendre Identity -- from Wolfram MathWorld. Calculus and Analysis. Special Functions. Elliptic Integrals.
Hace 4 días · 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 5 días · ϕ(x) = Jν(αx), which can be justified by direct substitution. For two distinct positive numbers k1 and k2, we consider two functions. ϕ1(x) = Jν(k1x) and ϕ2(x) = Jν(k2x). They are solutions of equations. d dx(x dϕ1(x) dx) + (k21x − ν x)ϕ1(x) = 0. and. d dx(x dϕ2(x) dx) + (k22x − ν x)ϕ2(x) = 0. respectively.
Hace 1 día · Quadratic Reciprocity (Legendre's statement). ... Gauss's fourth proof consists of proving this theorem (by comparing two formulas for the value of Gauss sums) and then restricting it to two primes. He then gives an example: Let a = 3, b = 5, c = 7, and d = 11.
Hace 5 días · Addition Formula for Cosine: cos ( a + b) = cos a cos b − sin a sin b. Subtraction Formula for Cosine: cos ( a − b) = cos a cos b + sin a sin b. Addition Formula for Sine: sin(a + b) = sina cosb + cosa sinb. Subtraction Formula for Sine: sin(a − b) = sina cosb − cosa sinb.
