"Spherical harmonics" Y(l,m) = (-1)^m sqrt((2 l + 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), (sin(theta)/2)^m sum(k, 0, l - m, (-1)^k (l + m + k)! / (l - m - k)! / (m + k)! / k! * ((1 - cos(theta)) / 2)^k)) "Verify equation (1)" D(f) = 1/sin(theta) d(sin(theta) d(f,theta), theta) + 1/sin(theta)^2 d(f,phi,2) for(l, 0, 2, for(m, -l, l, check(D(Y(l,m)) == -l (l + 1) Y(l,m)))) "ok" Run