Integrating Legendre Polynomials Pl & Pm

In summary, to integrate the expression involving Legendre polynomials, you can use the orthogonal properties listed in the Wikipedia article on Legendre polynomials and integration by parts. The orthogonality section provides the identity \displaystyle \frac{d}{dx}\left[(1-x^2)\frac{d}{dx}P(x)\right] = -\lambda P(x), where the eigenvalue \lambda corresponds to n(n+1). Knowing this property, you can use integration by parts to solve the problem successfully.
  • #1
Elliptic
33
0

Homework Statement


Integrate the expression
Pl and Pm are Legendre polynomials

Homework Equations






The Attempt at a Solution


Suppose that solution is equal to zero.
 

Attachments

  • Integrate.gif
    Integrate.gif
    1.8 KB · Views: 555
Physics news on Phys.org
  • #2
What properties do you know about Legendre Polynomials? If you can use the orthogonal properties that are listed in the article on Legendre polynomials in wikipedia, then integration by parts should do the trick.
 
  • #3
TheFurryGoat said:
What properties do you know about Legendre Polynomials? If you can use the orthogonal properties that are listed in the article on Legendre polynomials in wikipedia, then integration by parts should do the trick.

But, how make Pm'(x) I don't understand(recurrent differentiation formula?)
 
  • #4
Under the orthogonality section in the wikipedia article on Legendre polynomials, you find the identity
[itex]\displaystyle \frac{d}{dx}\left[(1-x^2)\frac{d}{dx}P(x)\right] = -\lambda P(x)[/itex]
where the eigenvalue [itex]\lambda[/itex] corresponds to [itex]n(n+1).[/itex]
I suppose [itex]P(x)=P_n(x)[/itex] for any [itex]n[/itex], but I'm not sure though. If this is the case, and you know this property, then integration by parts should do the trick.
 
  • #5
Thanks for help, I succeeded to do job.
 

What are Legendre Polynomials?

Legendre Polynomials, denoted as Pl(x), are a type of mathematical function used to represent solutions to certain differential equations. They are named after French mathematician Adrien-Marie Legendre and have many applications in physics, engineering, and other fields.

What is the purpose of integrating Legendre Polynomials?

The integration of Legendre Polynomials, denoted as Pm, is used to find the coefficients of a given function in terms of these polynomials. This allows for simplification and approximations of complex functions, making calculations easier and more accurate.

What is the difference between Pl and Pm?

The main difference between Pl and Pm is their degree or order. Pl has a degree of l, while Pm has a degree of m. This affects their shape and properties, and they are used for different purposes in integration.

How do you integrate Legendre Polynomials?

The integration of Legendre Polynomials involves using a combination of algebraic and trigonometric techniques to simplify the polynomial and solve for the unknown coefficients. The specific method may vary depending on the degree and order of the polynomial being integrated.

What are some applications of integrating Legendre Polynomials?

Integrating Legendre Polynomials has many applications in various fields, such as in calculating gravitational potential, electric potential, and magnetic field in physics. They are also used in solving differential equations, numerical analysis, and signal processing.

Similar threads

  • Calculus and Beyond Homework Help
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
2K
  • Calculus and Beyond Homework Help
Replies
18
Views
2K
  • Introductory Physics Homework Help
Replies
2
Views
760
  • Calculus and Beyond Homework Help
Replies
2
Views
2K
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Advanced Physics Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
4K
Back
Top