# Legendre Polynomial and Rodrigues' Formula

by kiwakwok
Tags: formula, legendre, polynomial, rodrigues
 P: 22 I am reading Jackson's electrodynamics book. When I went through the Legendre polynomial, I have a question. In the book, it stated that from the Rodrigues' formula we have Consider only the odd terms $\int_0^1dx\;P_l(x)=\left(-\frac{1}{2}\right)^{\frac{l-1}{2}}\frac{(l-2)!!}{2\left(\frac{l+1}{2}\right)!}$How to obtain this equation and how can I obtain the equation for even terms? Thanks in advance.
 Sci Advisor HW Helper P: 11,833 The general formula (irrespective of l even or odd) is http://www.wolframalpha.com/input/?i...[n%2Cz]+dz or set $\sigma=0$ in the formula (807) 7.126.1 in Gradshteyn-Ryzhik. The whole proof for l=odd is in Bell, W.W."Special Functions for Scientists and Engineers", (VanNostrand, 1967) as Example 2 on Page 86. You can make the proof by yourself with help of the fully solved case l=odd by making the necessary changes in the proof already given.
 Sci Advisor HW Helper P: 1,908 The formula is more easily derived using the generating function. The integral equals 1 for l=0, and is zero for all higher even l.

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