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Hace 5 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 · This function Q n (x) is called the Legendre function of the second kind. Mathematica has a build-in command for this function: LegendreQ[n, x]. Using Mathematica, we generate several Legendre polynomials of the second kind.
Hace 2 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 2 n − 1 or less on [−1, 1] .
Hace 5 días · Legendre's formula implies that the exponent of the prime = is always larger than the exponent for =, so each factor of five can be paired with a factor of two to produce one of these trailing zeros. The leading digits of the factorials are distributed according to Benford's law.
Hace 5 días · We investigate vertices for plane curves with singular points. As plane curves with singular points, we consider Legendre curves (respectively, Legendre immersions) in the unit tangent bundle over the Euclidean plane and frontals (respectively, fronts) in the Euclidean plane. We define a vertex using evolutes of frontals. After that we define a vertex of a frontal in the general case. It is ...
Hace 3 días · Bessel functions of the first kind, denoted as Jα(x), are solutions of Bessel's differential equation. For integer or positive α, Bessel functions of the first kind are finite at the origin ( x = 0 ); while for negative non-integer α, Bessel functions of the first kind diverge as x approaches zero.
Hace 3 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}
