What is the mathematical form of Legendre polynomial?
What is the mathematical form of Legendre polynomial?
Legendre Polynomials g ( x , t ) = ∑ n = 0 ∞ ( − 1 ) n t n 2 n n !
What is Airy’s equation?
The Airy equation is the second-order linear ordinary differential equation y″−xy=0. It occurred first in G.B. Airy’s research in optics [Ai]. Its general solution can be expressed in terms of Bessel functions of order ±1/3: y(x)=c1√xJ1/3(23ix3/2)+c2√xJ−1/3(23ix3/2).
What are Legendre polynomials used for?
For example, Legendre and Associate Legendre polynomials are widely used in the determination of wave functions of electrons in the orbits of an atom [3], [4] and in the determination of potential functions in the spherically symmetric geometry [5], etc.
Why do we use Rodrigues formula?
By extension, this can be used to transform all three basis vectors to compute a rotation matrix in SO(3), the group of all rotation matrices, from an axis–angle representation. …
What is the formula for the Legendre polynomials?
An especially compact expression for the Legendre polynomials is given by Rodrigues’ formula : P n ( x ) = 1 2 n n ! d n d x n ( x 2 − 1 ) n . {\\displaystyle P_ {n} (x)= {\\frac {1} {2^ {n}n!}} {\\frac {d^ {n}} {dx^ {n}}} (x^ {2}-1)^ {n}\\,.} ‘s. Among these are explicit representations such as
What is Legendre’s equation?
+n(n+1)y=0n>0, |x| <1 is known as Legendre’s equation. The general solution to this equation is given as a function of two Legendre functions as follows y=AP n(x)+BQ
What is the n(x) of the Legendre function?
n(x) are Legendre Functions of the first and second kind of order n. n(x) functions are called Legendre Polynomials or order n and are given by Rodrigue’s formula. n(x)= 1 2nn! n(x) can be used to obtain higher order polynomials.
Why do we use Legendre polynomials for gravitational potential?
The expression gives the gravitational potential associated to a point mass or the Coulomb potential associated to a point charge. The expansion using Legendre polynomials might be useful, for instance, when integrating this expression over a continuous mass or charge distribution.
