A multipole expansion is a series expansion of the effect produced by a given system in terms of an expansion parameter which becomes small as the distance away from the system increases. Therefore, the leading one or terms in a multipole expansion are generally the strongest. The first-order behavior of the system at large distances can therefore be obtained from the first terms of this series, which is generally much easier to compute than the general solution. Multipole expansions are most commonly used in problems involving the gravitational field of mass aggregations, the electric and magnetic fields of charge and current distributions, and the propagation of electromagnetic waves.
To compute one particular case of a multipole expansion, let R be the vector from a fixed reference point to the observation point, r be the vector from the reference point to a point in the body, and
(1)
be the vector from a point in the body to the observation point. The law of cosines then yields
(2)
where
(3)
so
(4)
Now define
(5)
(6)
then
(7)
But
is the generating function for Lendre polynomials as follows.
(8)
so
(9)
Any physical potential that obeys a
law can therefore be expressed as a multipole expansion 
(10)
The n = 0 term of this expansion, called the monopole term, can be pulled out by noting that
, so
(11)
The nth term
(12)
is commonly named according to the scheme summarized in the following table.
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A dipole consists of two point charges
located at 
, so the n = 0 terms vanishes. Now set up the coordinate system so that measures the angle from the charge-charge line with the midpoint of this line being the origin. Letting the observation point be 
and the density vector be 
, then the multipoles are given by
(13)
This gives
(14)
The first few cases are therefore
(15)
(16)
(17)
The multipole expansion can be rewritten using the spherical harmonic addition theorem
(18)
so
(19)
(20)
Define
(21)
which implies
(22)
then
(23)
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