Derivative of Spherical Harmonic for negative m

1. Mar 8, 2014

VVS

Hello!

1. The problem statement, all variables and given/known data
I want to evaluate the derivative of spherical harmonics with respect to the azimuthal angle and express it in terms of spherical harmonics.

2. Relevant equations and 3. The attempt at a solution
I have calculated the derivative of the spherical harmonic with respect to the azimuthal angle.
I know how to express the derivative of the spherical harmonic in terms of spherical harmonics for positive m. But I don't know how to do this for negative m.

View attachment Spherical_Harmonics.pdf

2. Mar 8, 2014

Jufro

Isn't the azimuthal angle the argument of the Exp[], in this case θ?

3. Mar 8, 2014

VVS

Yes you're right. I meant that I want the derivative wrt the polar angle.

4. Mar 8, 2014

Jufro

Hope this is what you were looking for.

It isn't the final answer because you can still express Cot ø and Exp(iθ) in terms of the spherical harmonics but that part is not the worst. May even be able to find them in a table somewhere.

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5. Mar 8, 2014

VVS

Hello!

Yes!! That is exactly what I was looking for.
Although I recalculated it and I get a slightly different result.
I will upload my calculations tomorrow or so. Right now I am too tired.

thanks

6. Mar 10, 2014