Legendre Polynomial and Rodrigues' Formula

[kiwakwok]

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.

 

[dextercioby]

The general formula (irrespective of l even or odd) is

http://www.wolframalpha.com/input/?i=Evaluate+integral+from+0+to+1+LegendreP[n,z]+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.

 

[Meir Achuz]

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.