sphericalHarmonic( l, m, θ, φ )

The spherical harmonic $$Y_l^m (\theta,\phi)$$ in Math. A solution of the partial differential equation

$\frac{ 1 }{ \sin \theta } \frac{ \partial }{ \partial \theta } \left( \sin \theta \frac{ \partial f }{ \partial \theta } \right) + \frac{ 1 }{ \sin^2 \theta } \frac{ \partial^2 f }{ \partial \phi^2 } + l(l+1) f = 0$

Related to associated Legendre polynomials by

$Y_l^m (\theta,\phi) = (-1)^m \sqrt{ \frac{ 2l+1 }{ 4\pi } \frac{ (l-m)! }{ (l+m)! } } e^{ im\phi } P_l^m (\cos\theta)$

Absolute value of the real part over real space:

Absolute value of the imaginary part over real space:

Absolute value over real space:

Related functions:   legendreP

Function category: orthogonal polynomials