Orthogonality Relationship for Legendre Polynomials

In summary, the basis {1,x,x²} is orthogonal but not orthonormal. The third basis function needs to be normalized using the integration formula $$\int_{-1}^1 dx (x^2 - 1/3)^2$$, and the first two basis functions should be normalized according to ##P_0(1) = \pm\frac{1}{\sqrt{2}}## and ##P_1(1) = \pm\sqrt{\frac{3}{2}}##. The Legendre polynomials are also orthogonal but not orthonormal over the interval [-1,1], so the orthonormal basis you construct from them may not be identical to the Legendre polynomials.
  • #1
LCSphysicist
646
162
Homework Statement
I am trying to orthogonalize {1,x,x²}
Relevant Equations
Just the inner product of functions space.
Suppose p = a + bx + cx².
I am trying to orthogonalize the basis {1,x,x²}
I finished finding {1,x,x²-(1/3)}, but this seems different from the second legendre polynomial.
1601451062834.png

What is the problem here? I thought could be the a problem about orthonormalization, but check and is not.
 
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  • #2
Your basis is orthogonal but not orthonormal. You need to compute the normalization for your third basis function, i.e. $$\int_{-1}^1 dx (x^2 - 1/3)^2$$.
 
  • #3
eys_physics said:
Your basis is orthogonal but not orthonormal. You need to compute the normalization for your third basis function, i.e. $$\int_{-1}^1 dx (x^2 - 1/3)^2$$.
Yeh, but as i said, i already do it. The integrate gives 8/45, taking the square and dividing by the module not get yet.
 
  • #4
Vendo seu perfil, acho que daria pra te responder em portugues :D
 
  • #5
Legendre polynomials are orthogonal but not orthonormal over the interval ##[-1,1]##. Thus, you shouldn't expect your orthonormal basis to be identical to the Legendre polynomials.

NB. If you are trying to construct and orthonormal set ##\{p_0,p_1,p_2\}## of polynomials over the interval ##[-1,1]## from the set ##\{1,x,x^2\}## of monomials. Then the first two are ##p_0(x) = \pm\frac{1}{\sqrt{2}}## and ##p_1(x) = \pm\sqrt{\frac{3}{2}}x## and not ##p_0(x) = 1## and ##p_1(x) = x##. As you can see, you have choice to make regarding the signs of ##p_0(x)## and ##p_1(x)##.
 
  • #6
The Legendre polynomials are usually normalized such that ##P_n(1) = 1##.
 

Related to Orthogonality Relationship for Legendre Polynomials

What is the orthogonality relationship for Legendre polynomials?

The orthogonality relationship for Legendre polynomials states that the integral of the product of two Legendre polynomials over a specific interval is equal to zero if the polynomials have different degrees, and is equal to a non-zero constant if the polynomials have the same degree.

What is the significance of the orthogonality relationship for Legendre polynomials?

The orthogonality relationship is significant because it allows for the decomposition of a function into a series of Legendre polynomials, making it easier to solve differential equations and other mathematical problems.

How is the orthogonality relationship used in practical applications?

The orthogonality relationship is used in many practical applications, such as in physics, engineering, and statistics. It is particularly useful in solving problems involving spherical harmonics, quantum mechanics, and Fourier series.

What are some properties of the orthogonality relationship for Legendre polynomials?

Some properties of the orthogonality relationship include the fact that it holds for any interval, not just the interval of -1 to 1, and that it can be extended to include complex-valued polynomials as well.

How is the orthogonality relationship related to the completeness of Legendre polynomials?

The orthogonality relationship is closely related to the completeness of Legendre polynomials, which means that any continuous function defined on the interval of -1 to 1 can be approximated by a series of Legendre polynomials. This relationship is important in many mathematical and scientific fields.

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