Legendre Polynomial and Rodrigues' Formulaby kiwakwok Tags: formula, legendre, polynomial, rodrigues 

#1
Oct3012, 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{l1}{2}}\frac{(l2)!!}{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. 



#2
Oct3012, 02:12 PM

Sci Advisor
HW Helper
P: 11,863

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 GradshteynRyzhik. 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. 



#3
Nov212, 09:00 PM

Sci Advisor
HW Helper
P: 1,935

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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