The representation of a function that depends on angles by a series. Usually, this is an infinite series of spherical harmonics. It is frequently possible to get a good approximation to a function by considering the first few terms. The first term is called the monopole term, the second the dipole term, the third the quadrupole term, the fourth the octupole term, etc.

Multipole expansions are used extensively to analyse electromagnetic and gravitational fields. In particular, multipole expansions of electromagnetic fields have important applications in nuclear theory and the theory of intermolecular forces.