Derivative of Spherical Harmonic for negative m

[VVS]

Hello!

Homework Statement

I want to evaluate the derivative of spherical harmonics with respect to the azimuthal angle and express it in terms of spherical harmonics.2. Homework 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.pdfThanks for your help.

 

[Jufro]

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

 

[VVS]

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

 

[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.

 

[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

 

[VVS]

Hey!

I got it now.

View attachment Spherical_Harmonics.pdf

If someone can point out mistakes, please do so.

VVS