Normalization coefficient for Spherical Harmonics with m=l

[david13579]

Homework Statement

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

Homework Equations

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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 don't 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.[/B]

 

[LmdL]

Hi,
They use a recurrence formula for sine integral:
$$\int sin^{2l+1} \left ( \theta \right )d\theta =-\frac{sin^{2l}\left ( \theta \right )cos\left ( \theta \right )}{2l+1}+\frac{2l}{2l+1}\int sin^{2l-1} \left ( \theta \right )d\theta$$
And since
$$sin\left ( 0 \right )=sin\left ( \pi \right )=0$$
You are left only with:
$$\int sin^{2l+1} \left ( \theta \right )d\theta =\frac{2l}{2l+1}\int sin^{2l-1} \left ( \theta \right )d\theta$$
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.