Legendre Polynomial and Rodrigues' Formula

  1. 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.
     
  2. jcsd
  3. dextercioby

    dextercioby 12,314
    Science Advisor
    Homework Helper

    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.
     
  4. Meir Achuz

    Meir Achuz 2,058
    Science Advisor
    Homework Helper
    Gold Member

    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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