# Multipole Expansion

A multipole expansion is a mathematical series representing a function that depends on angles — usually the two angles on a sphere. These series are useful because they can often be truncated, meaning that only the first few terms need to be retained for a good approximation to the original function. The function being expanded may be complex in general. Multipole expansions are very frequently used in the study of electromagnetic and gravitational fields, where the fields at distant points are given in terms of sources in a small region. The multipole expansion with angles is often combined with an expansion in radius. Such a combination gives an expansion describing a function throughout three-dimensional space.

The multipole expansion is expressed as a sum of terms with progressively finer angular features. For example, the initial term — called the zero-th, or monopole, moment — is a constant, independent of angle. The following term — the first, or dipole, moment — varies once from positive to negative around the sphere. Higher-order terms (like the quadrupole and octupole) vary more quickly with angles.

### Other articles related to "multipole expansion, expansions, expansion, multipole, multipole expansions":

Solutions To The Homogeneous Electromagnetic Wave Equation - Multipole Expansion
... The three-dimensional solutions of the Helmholtz Equation can be expressed as expansions in spherical harmonics with coefficients proportional to the ... However, applying this expansion to each vector component of E or B will give solutions that are not generically divergence-free (∇ · E = ∇ · B = 0), and therefore require additional ... The multipole expansion circumvents this difficulty by expanding not E or B, but r · E or r · B into spherical harmonics ...
Spherical Multipole Moments - Special Case of Axial Symmetry
... The spherical multipole expansion takes a simple form if the charge distribution is axially symmetric (i.e ... By carrying out the integrations that define and, it can be shown the multipole moments are all zero except when ... Using the mathematical identity the exterior multipole expansion becomes where the axially symmetric multipole moments are defined In the limit that the charge is confined to the -axis, we recover the exterior axial ...
Magnetic Monopole - Poles and Magnetism in Ordinary Matter
... Mathematically, the magnetic field of an object is often described in terms of a multipole expansion ... The first term in the expansion is called the "monopole" term, the second is called "dipole", then "quadrupole", then "octupole", and so on ... Any of these terms can be present in the multipole expansion of an electric field, for example ...
Distributed Multipole Analysis - Multipole Expansion
... of Cambridge University to describe the charge distribution of a molecule in terms of a multipole expansion around a number of centers ... The idea of using a multi-center multipole expansion was earlier proposed by Robert Rein ... A multipole series, consisting of a charge, dipole, quadrupole and higher terms is located at each center ...
Multipole Expansion - General Mathematical Properties
... Mathematically, multipole expansions are related to the underlying rotational symmetry of the physical laws and their associated differential equations ...

### Famous quotes containing the word expansion:

The fundamental steps of expansion that will open a person, over time, to the full flowering of his or her individuality are the same for both genders. But men and women are rarely in the same place struggling with the same questions at the same age.
Gail Sheehy (20th century)