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.
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 [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.
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.