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.

Read more about Multipole ExpansionExpansion in Spherical Harmonics, Applications of Multipole Expansions, Multipole Expansion of A Potential Outside An Electrostatic Charge Distribution, Interaction of Two Non-overlapping Charge Distributions, Examples of Multipole Expansions, General Mathematical Properties

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

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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
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