Maxwells equation for the static electric field with charge density
_{}
If we define the scalar potential in any of the usual ways
_{}
When _{} this is
Expressing the
_{}
This equation can be solved by separation of variables assuming a product of the
_{}
The solution for the angular part leads to the spherical harmonics _{}
If you assume no _{}dependence we get solutions in terms of _{} where _{}. These functions are Legendre polynomials. For general _{}dependence we find the associated Legendre polynomials _{} as shown above.
Wolfram
http://mathworld.wolfram.com/LegendrePolynomial.html
where the contour encloses the origin
and is traversed in a counterclockwise direction (Arfken
1985, p. 416).
The first few Legendre polynomials are



(2) 



(3) 



(4) 



(5) 



(6) 



(7) 



(8) 
Wolfram
http://mathworld.wolfram.com/LegendrePolynomial.html
http://en.wikipedia.org/wiki/Legendre_polynomials
In mathematics,
Legendre functions are solutions to Legendre's differential equation:
Legendre polynomials in multipole expansions
Figure 2
Legendre polynomials are also
useful in expanding functions of the form (this is the same as before, written
a little differently):
which arise naturally in multipole expansions. The lefthand side of the equation is the generating function for the Legendre
polynomials.
As an example, the electric potential Φ(r,θ) (in spherical coordinates) due to a point
charge located on the zaxis at z = a (Figure 2) varies like
If the radius r of the
observation point P is greater than a, the potential may be
expanded in the Legendre polynomials
where we have defined η = a/r < 1
and x = cos θ. This
expansion is used to develop the normal multipole expansion.
Conversely, if the radius r
of the observation point P is smaller than a, the potential may
still be expanded in the Legendre polynomials as above, but with a and r
exchanged. This expansion is the basis of interior multipole expansion.