Work out the normalization factor for the spherical harmonics, as follows. From Section 4.1.2 we know that

the problem is to determine the factor K^m_ℓ (which I quoted, but did not derive, in Equation 4.32). Use Equations 4.120, 4.121, and 4.130 to obtain a recursion relation giving K^m+1_ℓ in terms of K^m_ℓ. Solve it by induction on m to get K^m_ℓ up to an overall constant, C (ℓ) Ξ K⁰_ℓ. Finally, use the result of Problem 4.25 to fix the constant. You may find the following formula for the derivative of an associated Legendre function useful:

Equation 4.32

Equations 4.120

Equations 4.121

Y = Keimo pm (cos 9);