Lengendre polynomials

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In summary, Legendre polynomials are solutions to the differential equation (1-x^2)d^2y/dx^2 - 2x dy/dx+l(l+1)y=0, where l is an integer. The first five solutions are P0(x)=1, P1(x)=x, P2(x)=3/2x^2-1/2, P3(x)=5/2x^3-3/2x, P4(x)=35/8x^4-15/4x^2+3/8. To show that each polynomial Pl(x) solves the differential equation with its particular value l, we simply plug in the value of l for x and the
  • #1
ilikephysics
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I know that legendre polynomials are solutions of the differential equation is (1-x^2)d^2y/dx^2 - 2x dy/dx+l(l+1)y=0 where l is an integer. The first five solutions are P0(x)=1, P1(x)=x, P2(x)=3/2x^2-1/2, P3(x)=5/2x^3-3/2x, P4(x)=35/8x^4-15/4x^2+3/8

The problem is that I don't understand what the problem is telling me to do. It says to show that each of the polynomials Pl(x) solves the differentil equation with its particular value l. Do I just plug in l? For example, for P0(x)=1, would I plug in 1 for x and 0 for l? I'm really confused.


Another problem is that I have to show by doing 10 integrals that if l is not equal to m, that integral from -1 to 1 dxPl(x)Pm(x)=0 so that these polynomials are orthogonal on the interva1 [-1,1].

Do I just take a value for l and one for m 10 times. So for the first integral, m=1 and n=2?
 
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  • #2
[tex]y_l = P_l(x)[/tex]

is a solution to the differential equation

[tex](1-x^2)\frac{d^2y}{dx^2} - 2x \frac{dy}{dx} + l(l+1)y = 0[/tex]

They make up pairs. When l = 2, we have y_2 = P_2(x) is a solution to the differential equation given if 2 is plugged in for l (meaning the last term is 6).

If l = 15, then we have to plug in l = 15 into y_l and l = 15 into the differential equation. Then the solution y_15 will solve the differential equation made when we substitute 15 for l.

cookiemonster
 
  • #3
z solves an equation, f, if f(z)=0, o yes you just plug in the z which is P_l into the equation defining the l^th legendre polynomial.

secondly you must do the integrals for every pair of numbers (l,m) where l and m are one of 1,2,3,4
 

What are Legendre polynomials?

Legendre polynomials are a set of orthogonal polynomials named after French mathematician Adrien-Marie Legendre. They are used in many areas of physics and mathematics, particularly in solving differential equations.

What is the formula for Legendre polynomials?

The formula for Legendre polynomials is Pn(x) = (1/2nn!) * dn/dxn(x2 - 1)n, where n is the degree of the polynomial and dn/dxn represents the nth derivative.

What is the significance of Legendre polynomials?

Legendre polynomials have many important applications, such as in solving partial differential equations, approximating functions, and representing spherical harmonics. They are also used in physics to describe the behavior of particles in a potential field.

How are Legendre polynomials related to other orthogonal polynomials?

Legendre polynomials are a special case of Jacobi polynomials, which are a generalization of Legendre polynomials. They are also closely related to other orthogonal polynomials, such as Chebyshev polynomials and Hermite polynomials.

What are the properties of Legendre polynomials?

Legendre polynomials have many important properties, including orthogonality, recurrence relations, and explicit formulas for their coefficients. They also have a symmetric and bell-shaped distribution, and their roots are evenly spaced on the interval [-1, 1].

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