# Derivation: Normalization condition of Legendre polynomials

• schrodingerscat11
In summary, the orthonormality condition for Legendre polynomials can be derived from the Rodrigues formula using integration by parts. The key is to use the fundamental theorem of calculus to integrate the resulting expression.
schrodingerscat11
Greetings!

## Homework Statement

Starting from the Rodrigues formula, derive the orthonormality condition for the Legendre polynomials:
$\int^{+1}_{-1} P_l(x)P_{l'}(x)dx=(\frac{2}{2l + 1}) δ_{ll'}$

Hint: Use integration by parts

## Homework Equations

$P_l= \frac{1}{2^ll!}(\frac{d}{dx})^l (x^2-1)^l$ (Rodrigues formula)
∫udv = uv -∫vdu (integration by parts)

## The Attempt at a Solution

$\int^{+1}_{-1} P_l(x)P_{l'}(x)dx = \frac{1}{2^{l+l'}l!l'!} \int^{+1}_{-1} (\frac{d}{dx})^l \,(x^2-1)^l \, (\frac{d}{dx})^{l'} \,(x^2-1)^{l'}\,dx$

Integrating by parts:
∫udv = uv -∫vdu

Let $u = (\frac{d}{dx})^l (x^2-1)^l$
$\frac{du}{dx} = (\frac{d}{dx})^{l-1} (x^2-1)^{l-1}$
$du = (\frac{d}{dx})^{l-1} (x^2-1)^{l-1} dx$

Let $dv=(\frac{d}{dx})^{l'} (x^2-1)^{l'}dx$
$\int dv = \int (\frac{d}{dx})^{l'} (x^2-1)^{l'} dx$

Question: How do I integrate $\int (\frac{d}{dx})^{l'} (x^2-1)^{l'} dx$ ?

Thank you very much. :shy:

Think about the fundamental theorem of calculus
$$\int \mathrm{d} x f'(x)=f(x)+C$$
for a function with a continuous derivative!

1 person
Oooh... Yeah I remember. That was taught to us before. Thank you very much.

## 1. What is the normalization condition of Legendre polynomials?

The normalization condition of Legendre polynomials states that the integral of the square of a Legendre polynomial over the range [-1, 1] is equal to 1.

## 2. Why is the normalization condition important for Legendre polynomials?

The normalization condition is important because it ensures that the Legendre polynomials are properly scaled and have a maximum value of 1 at their peak. This allows for easier comparison and analysis of the polynomials.

## 3. How is the normalization condition of Legendre polynomials derived?

The normalization condition is derived by using the Gram-Schmidt process to orthogonalize the Legendre polynomials. This involves dividing each polynomial by its own norm and then multiplying by a normalization constant to ensure the integral of the squared polynomial is equal to 1.

## 4. Can the normalization condition of Legendre polynomials be generalized to other orthogonal polynomials?

Yes, the normalization condition can be generalized to other orthogonal polynomials by using the same process of dividing by the norm and multiplying by a normalization constant. However, the specific normalization constant will differ for each set of orthogonal polynomials.

## 5. What are some applications of the normalization condition for Legendre polynomials?

The normalization condition is used in many applications, including in physics and engineering for solving differential equations and in statistics for fitting data to a polynomial model. It is also used in numerical methods for approximating integrals and in signal processing for filtering and smoothing data.

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