"Verify Mathematica formula for derivative of spherical harmonic" DY(l,m) = m cos(theta) / sin(theta) Y(l,m) + sqrt((l - m) (l + m + 1)) exp(-i phi) Y(l,m + 1) Y(l,m) = (-1)^m sqrt((2l + 1) / (4 pi) (l - m)! / (l + m)!) * P(l,m) exp(i m phi) -- associated Legendre of cos theta (arxiv.org/abs/1805.12125) P(l,m,k) = test(m < 0, (-1)^m (l + m)! / (l - m)! P(l,-m), 2^(-m) sin(theta)^m sum(k,0,l - m, (-1)^k (l + m + k)! / (l - m - k)! / (m + k)! / k! * ((1 - cos(theta)) / 2)^k)) check(DY(0,0) == d(Y(0,0),theta)) check(DY(1,0) == d(Y(1,0),theta)) check(DY(1,1) == d(Y(1,1),theta)) check(DY(1,-1) == d(Y(1,-1),theta)) check(DY(2,0) == d(Y(2,0),theta)) check(DY(2,1) == d(Y(2,1),theta)) check(DY(2,2) == d(Y(2,2),theta)) check(DY(2,-1) == d(Y(2,-1),theta)) check(DY(2,-2) == d(Y(2,-2),theta)) "ok" Run