Proof of orthogonality of associated Legendre polynomial

In summary, the proof of orthogonality of associated Legendre polynomials is a mathematical demonstration that shows how these polynomials are orthogonal to each other when integrated over a specific range. This property is important in many applications, including solving problems in quantum mechanics and electrodynamics. The proof is derived using various mathematical techniques and has numerous applications in science and engineering. However, it has limitations in terms of applicable ranges, boundary conditions, and complexity for higher-order polynomials.
  • #1
MCKim
1
0
I want to prove orthogonality of associated Legendre polynomial.

In my textbook or many posts,
[tex]\int^{1}_{-1} P^{m}_{l}(x)P^{m}_{l'}(x)dx = 0 (l \neq l')[/tex]
is already proved.

But, for upper index m,
[tex]\int^{1}_{-1} P^{m}_{n}(x)P^{k}_{n}(x)\frac{dx}{ ( 1-x^{2} ) } = 0 (m \neq k)[/tex]
is not proved.

So, I tried to prove it using same method for the first case.
But I could not prove it.
Will anyone show me a hint or online reference?
 
Last edited:
Physics news on Phys.org
  • #2
plug in the formulas and do the integral. what have you done so far?
 

1. What is the proof of orthogonality of associated Legendre polynomials?

The proof of orthogonality of associated Legendre polynomials is a mathematical demonstration that shows how these polynomials are orthogonal to each other when integrated over a specific range. This property is essential in many applications, including solving problems in quantum mechanics and electrodynamics.

2. Why is the orthogonality of associated Legendre polynomials important?

The orthogonality of associated Legendre polynomials allows for the decomposition of functions into a series of these polynomials, which simplifies many mathematical calculations and makes them more manageable. It also helps in solving boundary value problems and finding solutions to differential equations.

3. How is the proof of orthogonality of associated Legendre polynomials derived?

The proof of orthogonality of associated Legendre polynomials is derived using various mathematical techniques, including integration by parts, the recurrence relation between Legendre polynomials, and the definition of orthogonal functions. It involves a series of steps and calculations, which are combined to show the orthogonality of these polynomials.

4. What are the applications of the proof of orthogonality of associated Legendre polynomials?

The proof of orthogonality of associated Legendre polynomials has numerous applications in various fields of science and engineering. It is used in solving boundary value problems in quantum mechanics, electrodynamics, and solid-state physics. It is also used in signal processing, image analysis, and computer graphics.

5. Are there any limitations to the proof of orthogonality of associated Legendre polynomials?

The proof of orthogonality of associated Legendre polynomials is valid only for certain ranges of values of the variables involved. It also assumes specific boundary conditions and may not hold for all types of functions. Additionally, the calculations involved in the proof can become complex for higher-order polynomials, making it challenging to apply in some cases.

Similar threads

I Determination of error in interpolating polynomial
    • Calculus
Replies
5
Views
384
I Legendre Differential Equation
    • Calculus
Replies
15
Views
3K
I Understanding Theorem 13 from Calculus 7th ed, R. Adams, C. Essex, 4.10
    • Calculus
Replies
10
Views
2K
I Why the integral of a differential does not give the function back in 2D?
    • Calculus
Replies
14
Views
1K
I Resolve the Recursion of Dickson polynomials
    • Calculus
Replies
1
Views
1K
I Legendre polynomials and Rodrigues' formula
    • Calculus
Replies
1
Views
3K
I Question about uniform convergence in a proof
    • Calculus
Replies
1
Views
154
Proof that the legendre polynomials are orthogonal polynomials
    • Calculus
Replies
3
Views
9K
Legendre Polynomials as an Orthogonal Basis
    • Introductory Physics Homework Help
Replies
2
Views
820
MHB Indefinite integral in division form
    • Calculus
Replies
1
Views
993

Hot Threads

Recent Insights

Back
Top