- #1

schrodingerscat11

- 89

- 1

## Homework Statement

Starting from the Rodrigues formula, derive the orthonormality condition for the Legendre polynomials:

[itex] \int^{+1}_{-1} P_l(x)P_{l'}(x)dx=(\frac{2}{2l + 1}) δ_{ll'} [/itex]

Hint: Use integration by parts

## Homework Equations

[itex]P_l= \frac{1}{2^ll!}(\frac{d}{dx})^l (x^2-1)^l[/itex] (Rodrigues formula)

∫udv = uv -∫vdu (integration by parts)

## The Attempt at a Solution

[itex] \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 [/itex]

Integrating by parts:

∫udv = uv -∫vdu

Let [itex] u = (\frac{d}{dx})^l (x^2-1)^l [/itex]

[itex] \frac{du}{dx} = (\frac{d}{dx})^{l-1} (x^2-1)^{l-1} [/itex]

[itex] du = (\frac{d}{dx})^{l-1} (x^2-1)^{l-1} dx[/itex]

Let [itex]dv=(\frac{d}{dx})^{l'} (x^2-1)^{l'}dx [/itex]

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

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

Thank you very much. :shy: