What is the Proof for the Relation Between Legendre Polynomials and Sums?

V, thank you for sharing this information with us. It's always great to have a reliable source for reference. In summary, the relation for the integral and sums involving Legendre polynomials can be found in the book "An elementary treatise on Fourier's series and spherical, cylindrical, and ellipsoidal harmonics" by W.E. Byerly. It can be located in article 106 of the book, which is available online at www.gutenberg.org.
  • #1
JiriV
2
0
Following relation seems to hold:

[itex]\int^{1}_{-1}\left(\sum \frac{b_{j}}{\sqrt{1-μ^{2}}} \frac{∂P_{j}(μ)}{∂μ}\right)^{2} dμ = 2\sum \frac{j(j+1)}{2j+1} b^{2}_{j}[/itex]

the sums are for j=0 to N and [itex]P_{j}(μ)[/itex] is a Legendre polynomial. I have tested this empirically and it seems correct.

Anyway, I would like to have i) either a proof, or ii) a reference in a book, by which it is obtained easily. Do you have some suggestions?

Thank you.
 
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  • #2
Hey JiriV and welcome to the forums.

I'm not too familiar with the legendre polynomial myself but I think I might be able to offer a few suggestions.

One suggestion is to expand the results of the integral and simplify. Another suggestion is to use the properties of the projection via an integral transform and a correct basis. For more of the specifics on this check out the following link and scroll down to the Gram-Schmidt process for getting the bases:

http://mathworld.wolfram.com/LegendrePolynomial.html

You might actually be better off doing the projection or using properties of the derivative in conjunction with the projection, but you would have to do a bit of investigation on your part.
 
  • #3
I will answer myself:

The relation above can be found in the book

W. E. Byerly, An elementary treatise on Fourier's series and spherical, cylindrical, and ellipsoidal harmonics, with applications to problems in mathematical physics. (Ginn & company, Boston, 1893).

(article 106). The book is available online at www.gutenberg.org.

Jiri
 

1. What are Legendre polynomials?

Legendre polynomials are a type of orthogonal polynomials named after French mathematician Adrien-Marie Legendre. They are used in mathematical analysis and physics, particularly in solving differential equations and representing functions on a bounded interval.

2. What is the significance of Legendre polynomials?

Legendre polynomials have many important applications in mathematics and physics. They are used in solving problems related to potential theory, quantum mechanics, and spherical harmonics. They also have applications in signal processing, image analysis, and data compression.

3. How are Legendre polynomials calculated?

Legendre polynomials can be calculated using the three-term recurrence relation or the Gram-Schmidt orthogonalization process. The three-term recurrence relation is a recursive formula that allows for the calculation of higher order Legendre polynomials from lower order ones. The Gram-Schmidt process involves a series of steps to construct an orthogonal basis from a set of linearly independent vectors.

4. What are the properties of Legendre polynomials?

Legendre polynomials have several important properties, including orthogonality, completeness, and symmetry. They are orthogonal with respect to a specific weight function on a given interval, and form a complete basis for square-integrable functions on that interval. They also exhibit symmetry about the origin, with odd-indexed polynomials being odd functions and even-indexed polynomials being even functions.

5. How are Legendre polynomials used in solving differential equations?

Legendre polynomials are often used in solving linear differential equations with constant coefficients. The solutions to these equations can be represented as a series of Legendre polynomials, allowing for the determination of specific coefficients. Additionally, the Legendre differential equation, which is a special case of the hypergeometric differential equation, has solutions in terms of Legendre polynomials.

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