Orthogonality of Legendre Polynomials

In summary, for spherical coordinates, we need to use Legendre Polynomials. In part a, the first 3 polynomials (P0(x), P1(x), and P2(x)) are graphed. In part b, the orthogonality relationship (eq 3.68) is evaluated, showing that these 3 functions are orthogonal to each other. In part c, the normalization result is shown to be 2/2l+1 as stated in eq 3.68. The equation with the theta terms is not necessary for this proof.
  • #1
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Homework Statement


For spherical coordinates, we will need to use Legendre Polynomials,
a.Sketch graphs of the first 3 – P0(x), P1(x), and P2(x).

b.Evaluate the orthogonality relationship (eq 3.68) to show these 3 functions are
orthogonal to each other. (3 integrals).

c.Show that the normalization result is 2/2l+1 as stated in eq 3.68.


Homework Equations





The Attempt at a Solution


I believe I got part a, but I am unsure how to do the rest. I don't see where the equation that has the theta terms, comes into use.
 

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  • #2
The equation with the theta terms is irrelevant. It is there to show the main application, namely Spherical Harmonics. You just need to carry out the integration to show you get 0 for distinct l,l' and 1 (with the given normalization) if l = l'.
 
  • #3
Ahh ok I was wondering this. I realized maybe this was the case a few minutes ago, and tried to work it out. Here is what I just did. I think this is correct... Any thoughts are appreciated.
 

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FAQ: Orthogonality of Legendre Polynomials

What is the definition of orthogonality in the context of Legendre Polynomials?

Orthogonality in the context of Legendre Polynomials refers to the property of being perpendicular or independent of each other in a mathematical sense. In other words, when two Legendre Polynomials are multiplied together and integrated over the range of -1 to 1, the result is equal to 0, indicating that they are orthogonal to each other.

Why is orthogonality important in the study of Legendre Polynomials?

Orthogonality is important in the study of Legendre Polynomials because it allows us to easily manipulate and solve equations involving these polynomials. By using the orthogonality property, we can simplify integrals and solve differential equations, making calculations and analysis more efficient.

How are the orthogonality properties of Legendre Polynomials used in real-world applications?

The orthogonality properties of Legendre Polynomials are used in a variety of real-world applications, such as in physics, engineering, and statistics. For example, they are used to model and analyze physical phenomena, such as the motion of particles and the behavior of electromagnetic fields. They are also used in signal processing and data analysis to extract useful information from noisy data sets.

Can two Legendre Polynomials with different orders be orthogonal to each other?

Yes, two Legendre Polynomials with different orders can be orthogonal to each other. The orthogonality of Legendre Polynomials is based on the integration of their product over a specific range, not on their orders. Therefore, even if two polynomials have different orders, they can still be orthogonal if their product integrates to 0 over the given range.

Are Legendre Polynomials the only set of orthogonal polynomials?

No, Legendre Polynomials are not the only set of orthogonal polynomials. There are many other families of orthogonal polynomials, such as Chebyshev Polynomials, Hermite Polynomials, and Laguerre Polynomials. Each set of orthogonal polynomials has its own unique properties and applications in various fields of study.

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