Proving Orthogonality of Legendre Polynomials P3 and P1

In summary, to show that two Legendre polynomials (Pn and Pm) are orthogonal, you need to use the following test: integrate Pn(x)Pm(x) from -1 to 1 and the result should be 0. To prove that P3 and P1 are orthogonal, you would need to use this formula. "Orthogonal" in this case means that the two polynomials are perpendicular to each other. You can also use the Rodriguez formula to find the Legendre polynomial in three dimensions. The rule for obtaining the general Pn Legendre Polynomial in three dimensions is to integrate Pn(x)Pm(x) over the entire range.
  • #1
stunner5000pt
1,461
2
To show that two Legendre polynomials(Pn and Pm) are orthogonal wht is the test that i have to use?

is it this?
[tex] \int_{-1}^{1} P_{n}(x)P_{m}(x) dx = 0 [/tex]

in that case to prove that P3 and P1 are orthogonal i have to use the above formula??
 
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  • #2
What is the definition of "orthogonal" in this case?
 
  • #3
you can also use rodriguez formula for this
 
  • #4
legendre polynom in three dimension

Hello,
does anyone know the rule to obtain a general [tex]P^n[/tex] Legendre Polynomial in three dimension?

thanks!
 
  • #5
sorry, I wronged:the rule to obtain the general [tex]P_n[/tex] Legendre Polynomial
 

FAQ: Proving Orthogonality of Legendre Polynomials P3 and P1

What is the definition of orthogonality for Legendre polynomials P3 and P1?

The orthogonality of Legendre polynomials P3 and P1 refers to the property that the integral of their product over a specific range is equal to zero, indicating that the two polynomials are perpendicular to each other in a mathematical space.

How can I prove the orthogonality of Legendre polynomials P3 and P1?

The orthogonality of Legendre polynomials P3 and P1 can be proven by using the definition of orthogonality and evaluating the integral of their product over the range of interest. This integral can be simplified using integration by parts or other mathematical techniques to show that it equals zero.

Why is it important to prove the orthogonality of Legendre polynomials P3 and P1?

Proving the orthogonality of Legendre polynomials P3 and P1 is important because it allows for the use of these polynomials in various mathematical methods and applications. For example, in physics and engineering, Legendre polynomials are commonly used to represent the solutions to differential equations, and their orthogonality is crucial for these solutions to be accurate.

Are there any other methods for proving the orthogonality of Legendre polynomials P3 and P1?

Yes, there are other methods for proving the orthogonality of Legendre polynomials P3 and P1. One common method is using the Rodrigues formula, which expresses Legendre polynomials in terms of derivatives of a generating function. This method can be useful for proving the orthogonality of Legendre polynomials of higher orders as well.

Can the orthogonality of Legendre polynomials P3 and P1 be generalized to other sets of polynomials?

Yes, the concept of orthogonality can be extended to other sets of polynomials, such as Chebyshev polynomials and Hermite polynomials. However, the specific conditions and methods for proving orthogonality may vary for each set of polynomials.

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