Legendre Polynomial and Rodrigues' Formula

by kiwakwok
Tags: formula, legendre, polynomial, rodrigues
kiwakwok is offline
Oct30-12, 08:21 AM
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
How to obtain this equation and how can I obtain the equation for even terms?

Thanks in advance.
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dextercioby is offline
Oct30-12, 02:12 PM
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The general formula (irrespective of l even or odd) is


or set [itex] \sigma=0[/itex] 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.
Meir Achuz
Meir Achuz is offline
Nov2-12, 09:00 PM
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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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