Question

8. Express the polynomial 2x' +x+3 in terms of Legendre's polynomials.​

Answer 1

The Legendre's polynomial is, $3p_{1} (x)+3p_{0} (x)$.

• Legendre polynomials in physical science and mathematics are a system of complete and orthogonal polynomials with a wide range of mathematical properties and multiple applications. They are named after Adrien-Marie Legendre, who made their discovery in 1782. There are numerous ways to define them, and each explanation highlights a particular characteristic while also offering generalisations, linkages to various mathematical structures, and uses in both physical and numerical applications. Legendre functions, Legendre polynomials, Legendre functions of the second order, and associated Legendre functions are closely connected to the Legendre polynomials..
• Legendre's differential equation naturally occurs in physical contexts anytime Laplace's equation (and similar partial differential equations) are solved by separating the variables in spherical coordinates. In this sense, the Legendre polynomials are the subset of the spherical harmonics that is left invariant by rotations about the polar axis. These are the eigenfunctions of the angular component of the Laplacian operator.

Here, according to the given information, the polynomial is given as,

$2x^{1}+x + 3$ , or, 3x + 3.

Now, let 3x + 3 = Cx + D.

Comparing the coefficients on both sides, we get,

C = 3 and D = 3.

Hence, the Legendre's polynomial is, $3p_{1} (x)+3p_{0} (x)$.

Learn more here

https://brainly.in/question/32910217

#SPJ1