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Hace 3 días · We can iterate and show that each coefficient is either proportional to a0 or a1. However, for n an integer, sooner, or later, m = n and the series truncates. am = 0 for m > n. Thus, we obtain polynomial solutions. These polynomial solutions are the Legendre polynomials, which we designate as y(x) = Pn(x).
Hace 4 días · which is exact for polynomials of degree 2n − 1 or less. 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 ...
Hace 3 días · Legendre Identity -- from Wolfram MathWorld. Calculus and Analysis. Special Functions. Elliptic Integrals.
Hace 5 días · Suppose that f is an odd function on interval [−1, 1]. Since P n (x) is odd when n is odd and P n (x) is even when n is even, then the Legendre coefficients of f with even indices are all zero (c 2j = 0). The Legendre series of f contains only odd indexed polynomials. Similarly, if f is an even function, then its Legendre series contains only even indexed polynomials.
Hace 1 día · Continued fraction. A finite regular continued fraction, where is a non-negative integer, is an integer, and is a positive integer, for . In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this ...
Hace 3 días · Laguerre polynomials. In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are nontrivial solutions of Laguerre's differential equation: which is a second-order linear differential equation. This equation has nonsingular solutions only if n is a non-negative integer.
Hace 4 días · 1. 2. We have introduced framed surfaces as smooth surfaces with singular points. The framed surface is a surface with a moving frame based on the unit normal vector of the surface. Thus, the notion of framed surfaces (respectively, framed base surfaces) is locally equivalent to the notion of Legendre surfaces (respectively, frontals).
