# Derivation: Normalization condition of Legendre polynomials

1. Feb 10, 2014

### physicsjn

Greetings!

1. The problem statement, all variables and given/known data
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

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

3. 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:

2. Feb 10, 2014

### vanhees71

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

3. Feb 11, 2014

### physicsjn

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