Which of the following is a Legendre equation?
The Legendre differential equation is the second order ordinary differential equation (ODE) which can be written as: ( 1 − x 2 ) d 2 y / d x 2 − 2 x d y / d x + l ( l + 1 ) y = 0 {\displaystyle (1-x^{2})d^{2}y/dx^{2}-2xdy/dx+l(l+1)y=0\,}
What is Legendre function 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.
What is the difference between Legendre function and Legendre polynomial?
Definition via differential equation
. A two-parameter generalization of (Eq. 1) is called Legendre’s general differential equation, solved by the Associated Legendre polynomials. Legendre functions are solutions of Legendre’s differential equation (generalized or not) with non-integer parameters.
What is Legendre function of the first kind?
real, the Legendre function of the first kind simplifies to a polynomial, called the Legendre polynomial. The associated Legendre function of first kind is given by the Wolfram Language command LegendreP[n, m, z], and the unassociated function by LegendreP[n, z].
What is the use of Legendre differential equation explain with example?
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 linear 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 are properties of Legendre polynomial?
11.2 Properties of Legendre polynomials
Let F(x t) be a function of the two variables x and t that can be expressed as a Taylor’s series in t, ∑ncn(x)tn. The function F is then called a generating function of the functions cn.
What is the generating function for Legendre polynomials?
The Legendre polynomials can be alternatively given by the generating function ( 1 − 2 x z + z 2 ) − 1 / 2 = ∑ n = 0 ∞ P n ( x ) z n , but there are other generating functions.
How do you find a Legendre polynomial?
Legendre’s ODE II: Deriving a formula for Legendre Polynomials – YouTube
What is the generating function of Legendre polynomial?
What is the ordinary point of Legendre differential equation?
(x+1)(x-1) Stence the Legendre Equation has regular singular points at x=±1. 1.
What is the solution of Legendre differential equation?
Who discovered Legendre polynomial?
| Adrien-Marie Legendre | |
|---|---|
| Alma mater | Collège Mazarin |
| Known for | Associated Legendre polynomials Legendre transformation Legendre polynomials Legendre transform Elliptic functions Introducing the character ∂ |
| Scientific career | |
| Fields | Mathematician |
What is the generating function of Legendre polynomials justify why it is called so?
Generating Function for Legendre’s Polynomial Pn(x) The function (1−2x h + h. 2. )−1/2 is called as the generating function for. Pn(x) and, therefore, Pn(x) is the coefficient of h.
How do you solve a Legendre function?
Solving ODEs by Series Solutions: Legendre’s ODE – YouTube
Are Legendre polynomials even?
They are solutions to a very important differential equation, the Legendre equation: The polynomials may be denoted by Pn(x) , called the Legendre polynomial of order n. The polynomials are either even or odd functions of x for even or odd orders n.