Spherical Harmonics: Proving Y_L^M(0,phi)

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Homework Statement

Prove that

{Y_{L}^{M}\left ( 0,\varphi \right )=\left ( \frac{2L+1}{4\pi } \right )^{1/2}\delta _{M,0}

Homework Equations

Y_{L}^{M}\left ( \theta,\varphi \right )=\left ( \frac{(2L+1)(L-M)!}{4\pi(L+M)! } \right )^{1/2}P_{L}^{M}(cos\theta )e^{im\varphi }

\int_{\varphi =0}^{2\pi }\int_{\theta =0}^{\pi }Y_{L1}^{M1}\left ( \theta ,\varphi \right )Y_{L2}^{M2}\left ( \theta,\varphi \right )sin\theta d\theta d\varphi = \delta _{N1,N2}\delta _{M1,M2}

The Attempt at a Solution

I think i need integrate/combine relevant equations in first equation,but ...?

 

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written in latex

 

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I have proved the claim. Task completed.