Spherical Harmonics
A set of special orthogonal functions defined on the 2D sphere surface serving as angular solutions to Laplace's equation in 3D spherical coordinates.
Governing formula
Y_l,m(θ, φ) = √(((2l+1)/(4π)) ((l-m)!/(l+m)!)) P_l^m(cos θ) e^(i m φ)
SI unit
Spherical Surface Functions
In depth
Form a complete orthogonal basis for functions defined on a sphere. Used across quantum mechanics (atomic orbital angular shapes) and astrophysics (CMB anisotropy maps).
Examples in the real world
Decomposing Planck space telescope Cosmic Microwave Background temperature maps into angular power spectrum multipoles C_l.