Legendre Polynomial and Rodrigues' Formula


by kiwakwok
Tags: formula, legendre, polynomial, rodrigues
kiwakwok
kiwakwok is offline
#1
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
[itex]\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)!}[/itex]
How to obtain this equation and how can I obtain the equation for even terms?

Thanks in advance.
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dextercioby
dextercioby is offline
#2
Oct30-12, 02:12 PM
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The general formula (irrespective of l even or odd) is

http://www.wolframalpha.com/input/?i...[n%2Cz]+dz

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
#3
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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