Orthogonality of Legendre Polynomials from Jackson

In summary, Jackson starts with the following orthogonality statement and jumps in his proof to Equation 3.18. He mentions integration by parts on the "first term" but I don't see how he gets to Equation 3.17.
  • #1
Demon117
165
1
Hello all!

I am trying to work through and understand the derivation of the Legendre Polynomials from Jackson's Classical electrodynamics. I have reached a part that I cannot get through however. Jackson starts with the following orthogonality statement and jumps (as it seems) in his proof:

Equation 3.17 states:

[itex]\int P_{l'}[\frac{d}{dx} ([1-x^{2}]\frac{dP_{l}}{dx})+l(l+1)P_{l}(x)]dx=0[/itex]

He mentions integration by parts on the "first term" but I don't see how he gets to

Equation 3.18:

[itex]\int [(x^{2}-1)\frac{dP_{l}}{dx} \frac{dP_{l'}}{dx} +l(l+1)(P_{l'}(x)P_{l}(x))]dx=0[/itex]

I don't see this. Could someone please explain or give a hint to the intermediate step here? I'm afraid I just do not see it. Thanks.
 
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  • #2
You understand integration by parts, yes? To wit,

[tex] \int udv = uv-\int vdu [/tex]

So by the chain rule we can also say

[tex] \int u \frac{dv}{dx}dx = uv-\int v \frac{du}{dx}dx [/tex]

So it is easy to see that [itex]u=P_{\ell'}[/itex] and [itex]v=\left[ \left(1-x^2\right)\frac{dP_\ell}{dx}\right][/itex]. In this manner we arrive at

[tex] \int_{-1}^1 P_{\ell'} \frac{d}{dx} \left[ \left(1-x^2\right)\frac{dP_\ell}{dx}\right] dx = \left. P_{\ell'} \left[ \left(1-x^2\right)\frac{dP_\ell}{dx}\right] \right|^{x=1}_{x=-1} - \int_{-1}^1 \left[ \left(1-x^2\right)\frac{dP_\ell}{dx}\right] \frac{d P_{\ell'}}{dx} dx [/tex]

The first term works out to be zero obviously and thus we say that,

[tex] \int_{-1}^1 P_{\ell'} \frac{d}{dx} \left[ \left(1-x^2\right)\frac{dP_\ell}{dx}\right] dx = \int_{-1}^1 \left[ \left(x^2-1\right)\frac{dP_\ell}{dx}\right] \frac{d P_{\ell'}}{dx} dx [/tex]
 
  • #3
Born2bwire said:
You understand integration by parts, yes? To wit,

[tex] \int udv = uv-\int vdu [/tex]

So by the chain rule we can also say

[tex] \int u \frac{dv}{dx}dx = uv-\int v \frac{du}{dx}dx [/tex]

So it is easy to see that [itex]u=P_{\ell'}[/itex] and [itex]v=\left[ \left(1-x^2\right)\frac{dP_\ell}{dx}\right][/itex]. In this manner we arrive at

[tex] \int_{-1}^1 P_{\ell'} \frac{d}{dx} \left[ \left(1-x^2\right)\frac{dP_\ell}{dx}\right] dx = \left. P_{\ell'} \left[ \left(1-x^2\right)\frac{dP_\ell}{dx}\right] \right|^{x=1}_{x=-1} - \int_{-1}^1 \left[ \left(1-x^2\right)\frac{dP_\ell}{dx}\right] \frac{d P_{\ell'}}{dx} dx [/tex]

The first term works out to be zero obviously and thus we say that,

[tex] \int_{-1}^1 P_{\ell'} \frac{d}{dx} \left[ \left(1-x^2\right)\frac{dP_\ell}{dx}\right] dx = \int_{-1}^1 \left[ \left(x^2-1\right)\frac{dP_\ell}{dx}\right] \frac{d P_{\ell'}}{dx} dx [/tex]

Of course I understand Integration by parts, and in fact I just barely started it myself so this whole conversation is null. Thank you for your time, but I just had an aha! moment :) Sorry about that.
 

Related to Orthogonality of Legendre Polynomials from Jackson

1. What is the significance of the orthogonality of Legendre polynomials?

The orthogonality of Legendre polynomials is a fundamental property that makes them useful in many mathematical and scientific applications. It means that when two different polynomials are multiplied together and integrated over a certain interval, the result is zero. This property allows for efficient calculations and simplification of complex equations.

2. How are Legendre polynomials related to spherical harmonics?

Legendre polynomials are closely related to spherical harmonics, which are functions used to describe the spatial distribution of a scalar field in three-dimensional space. The spherical harmonics are constructed using the Legendre polynomials and have applications in quantum mechanics, electromagnetism, and other fields of physics.

3. What is the mathematical formula for Legendre polynomials?

The mathematical formula for Legendre polynomials is Pn(x) = (1/2nn!) * (d/dx)n[(x2-1)n]. This formula generates a sequence of polynomials with increasing degree (n) and can be used to calculate the specific polynomial for any given value of n and x.

4. Can Legendre polynomials only be used for solving differential equations?

No, Legendre polynomials have many applications beyond solving differential equations. They are also used in approximation theory, numerical analysis, and signal processing. They have been applied to diverse problems, such as predicting stock market trends and modeling chaotic systems.

5. How are Legendre polynomials related to other families of orthogonal polynomials?

Legendre polynomials are a special case of a larger family of orthogonal polynomials known as the classical orthogonal polynomials. This family also includes Chebyshev polynomials, Hermite polynomials, and Laguerre polynomials. Legendre polynomials have the unique property of being orthogonal over the interval [-1,1], while other families have different intervals of orthogonality.

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