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I was recently reading Barton's book.

I reached the part where he proved that in spherical polar coordinates

##δ(\vec r - \vec r')=1/r^2δ(r-r')δ(cosθ-cosθ')δ(φ-φ')##

##=1/r^2δ(r-r')δ(\Omega -\Omega')##

Then he said that the most fruitful presentation of ##δ(\Omega-\Omega')## stems from the closure property of the spherical harmonics ##Y_{lm}(\Omega)## which constitute a complete orthonormal set over the surface of the unit sphere.

Then he said that the spherical harmonics satisfies the remarkable addition theorem:

##\sum_{m=-l}^{m=l}Y_{lm}^*(\Omega')Y_{lm}(\Omega)=(2l+1/4π) P_l(\vec r \cdot \vec r')##

My problem is that I didn't get from where he obtained this relation.

Besides he said that the angle##\chi## between ##\vec r## and ##\vec r'## is

##cos\chi=\vec r \cdot \vec r'={cosθcosθ'+sinθsinθ'cos(Φ-Φ')}##

Where the vector where ##\vec r## and ##\vec r'## are two vectors in spherical coordinates.

If somebody can help me obtaining this relation too.

Thanks.

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# A Spherical harmonics

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