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Normalization coefficient for Spherical Harmonics with m=l

  1. Dec 15, 2014 #1
    1. The problem statement, all variables and given/known data
    Well it is not the problem itself that bothers me but the maths behind a part of it. As part of finding the coefficient I had to solve the integral of (Sin(x))^(2l+ 1). The solution given by the solution manual just pretty much jumps to the final answer http://i.imgur.com/hhoeLKE.png

    2. Relevant equations
    ...

    3. The attempt at a solution
    Using substitution and the binomial theorem I was able to get a solution (of the integral part only, It would still need solving for the coefficient) and got this http://i.imgur.com/wnOUNIp.png

    The problem is I dont see how to get from my answer to the one on the solution manual. I know they are equivalent because I checked numerically for different values of l.
     
  2. jcsd
  3. Dec 15, 2014 #2
    Hi,
    They use a recurrence formula for sine integral:
    [tex]\int sin^{2l+1} \left ( \theta \right )d\theta =-\frac{sin^{2l}\left ( \theta \right )cos\left ( \theta \right )}{2l+1}+\frac{2l}{2l+1}\int sin^{2l-1} \left ( \theta \right )d\theta[/tex]
    And since
    [tex]sin\left ( 0 \right )=sin\left ( \pi \right )=0[/tex]
    You are left only with:
    [tex]\int sin^{2l+1} \left ( \theta \right )d\theta =\frac{2l}{2l+1}\int sin^{2l-1} \left ( \theta \right )d\theta[/tex]
    Finally they just sort of "calculate" this recurrently, and you are left only with the product of coefficients on the right side before the integral.
     
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