Computing Spherical Harmonics

[~Sam~]

Homework Statement

For the spherical harmonics U_mn; V_mn, compute the ones of orders
0,1, 2.

U_mn=cos(nθ)sinⁿ($\varphi$)P_mⁿ(cos($\varphi$))
V_mn=sin(nθ)sinⁿ($\varphi$)P_mⁿ(cos($\varphi$))
(b) How many non-zero spherical harmonics are there of order k?

Homework Equations

Equations of U_mn; V_mn given

The Attempt at a Solution

What I'm confused about is that these aren't the usual spherical harmonics that I see, plus the notation is different. Is it the different notation? And shouldn't spherical harmonics have 2k+1 for part b?

 

[mighty2000]

Thank you for your question. I believe the notation used in this problem is simply different from what you are used to seeing. The equations for Umn and Vmn are indeed the usual spherical harmonics, just written in a slightly different form. To compute the ones of orders 0, 1, and 2, you can simply substitute the given values for n and θ into the equations and evaluate them.

As for part b, the number of non-zero spherical harmonics of order k is indeed 2k+1. I believe this may have been a typo in the original question. I apologize for any confusion this may have caused. If you have any further questions or need clarification, please don't hesitate to ask.