What is the Rodrigues formula for Legendre polynomial?
Some very elegant representations of special functions are possible with use of contour integrals in the complex plane. Recall Rodrigues’ formula for Legendre polynomials (13.78): (14.72) d ℓ dx ℓ ( x 2 – 1 ) ℓ .
What is the purpose of Rodrigues formula?
We can construct an integral representation of Legendre polynomials with the help of Rodrigues’ formula. The representation we obtain this way is called the Schlaefli integral. The integral representation of a function is an expression involving a contour integral.
What is Legendre polynomial in physics?
The Legendre polynomials are solutions of this and related Equations that appear in the study of the vibrations of a solid sphere (spherical harmonics) and in the solution of the Schrödinger Equation for hydrogen-like atoms, and they play a large role in quantum mechanics.
What is hermite differential equation?
where is a constant is known as Hermite differential equation. When is an. odd integer i.e., when = 2 + 1; = 0,1,2 … …. then one of the solutions of. equation (1) becomes a polynomial.
Are Legendre polynomials even?
One of the varieties of special functions which are encountered in the solution of physical problems is the class of functions called Legendre polynomials. They are solutions to a very important differential equation, the Legendre equation: The polynomials are either even or odd functions of x for even or odd orders n.
What is the use of Legendre differential equation?
Legendre‟s equation occur in many areas of applied mathematics, physics and chemistry in physical situation with a spherical geometry such as flow of an ideal fluid past a sphere, the determination of the electric field due to a charged sphere and the determination of the temperature distribution in a sphere given its …
What is Legendre differential equation?
Since the Legendre differential equation is a second-order ordinary differential equation, it has two linearly independent solutions. A solution which is regular at finite points is called a Legendre function of the first kind, while a solution which is singular at is called a Legendre function of the second kind.
What is the formula for rotation?
Rotation Formula:
| Rotation | Point coordinate | Point coordinate after Rotation |
|---|---|---|
| Rotation of 90^{0} (Anti-Clockwise) | (x, y) | (-y, x) |
| Rotation of 180^{0} (Both) | (x, y) | (-x, -y) |
| Rotation of 270^{0} (Clockwise) | (x, y) | (-y, x) |
| Rotation of 270^{0} (Anti-Clockwise) | (x, y) | (y, -x) |
What is the general formula of Legendre’s polynomial s?
(The general formula of Legendre Polynomial s is given by following equation: Pk(x) = k 2 k − 1 2 ∑ m = 0 (− 1)m(2k − 2m)! 2km!(k − m)! 1 (k − 2m)!xk − 2m The Rodrigues’ formula is: 1 2kk! dk dxk[(x2 − 1)k]
What is Rodrigues’ formula for the Legendre function?
This is Rodrigues’ formula for the Legendre function. By means of the binominal formula we get (28) The summation starts at r=0 and end when n-2ris 0 (n=even) or 1 (n=odd). In evaluating Pnit is most easy to use (27) directly. Thus, .
Is there a solution to Legendre’s differential equation?
Legendre Polynomials: Rodriques’ Formula and Recursion Relations Jackson says “By manipulation of the power series solutions it is possible to obtain a compact representation of the Legendre polynomials known as Rodrigues’ formula.” Here is a proof that Rodrigues’ formula indeed produces a solution to Legendre’s differential equation.
How do you find the Legendre differential equation with derivatives?
A more intuitive approach is to start at the polynomials y ( x) = ( 1 − x 2) n. and take derivates, and verifty that the derivatives taken n times will get you to the Legendre differential equation. That is, we have that ( 1 − x 2) y ′ + 2 n x y = 0.