written as where the represent the components of a unit vector in the direction given by the angles and , and indices are implicitly summed. Here, the term is the monopole; is a set of three numbers representing the dipole; and so on. In the above expansions, the coefficients may be real or complex. If the function being expressed as a multipole expansion is real, however, the coefficients must satisfy certain properties. In the spherical harmonic expansion, we must have In the multi-vector expansion, each coefficient must be real: While expansions of scalar functions are by far the most common application of multipole expansions, they may also be generalized to describe tensors of arbitrary rank. This finds use in multipole expansions of the vector potential in electromagnetism, or the metric perturbation in the description of gravitational waves. For describing functions of three dimensions, away from the coordinate origin, the coefficients of the multipole expansion can be written as functions of the distance to the origin, —most frequently, as a Laurent series in powers of . For example, to describe the electromagnetic potential, , from a source in a small region near the origin, the coefficients may be written as: Applications Multipole expansions are widely used in problems involving gravitational fields of systems of masses, electric and magnetic fields of charge and current distributions, and the propagation of electromagnetic waves. A classic example is the calculation of the exterior multipole moments of atomic nuclei from their interaction energies with the interior multipoles of the electronic orbitals. The multipole moments of the nuclei report on the distribution of charges within the nucleus and, thus, on the shape of the nucleus. Truncation of the multipole expansion to its first non-zero term is often useful for theoretical calculations. Multipole expansions are also useful in numerical simulations, and form the basis of the fast multipole method of Greengard and Rokhlin, a general technique for efficient computation of energies and forces in systems of interacting particles. The basic idea is to decompose the particles into groups; particles within a group interact normally (i.e., by the full potential), whereas the energies and forces between groups of particles are calculated from their multipole moments. The efficiency of the fast multipole method is generally similar to that of Ewald summation, but is superior if the particles are clustered, i.e. the system has large density fluctuations. Multipole expansion of a