# Laplace's Equation, Legendre's Equation, Legendre Polynomials

## Problem statement

In the theoretical development of electric fields in electrostatic systems, one begins
with solutions of Laplace's equation,

*d*^{2}U/dx^{2} + d^{2}U/dy^{2} + d^{2}U/dz^{2}
= 0 Equ (2)

(Reference: J. R. Reitz and F. J. Milford, Foundations of Electromagnetic Theory
[Addison-wesley, 1960]). If the boundary conditions are adapted to spherical symmetry, as
is often the case, the equation is written in spherical polar coordinates. Then special
solutions are sought by separating the radial and angular coordinates (*r*, *q*, and *f*). For the q-equation one obtains the Legendre DE:

*d/dz [ (1-z*^{2}) dZ/dz ] - (m^{2}/(1-z^{2}))Z
+ n(n+1) Z = 0.

in which *z = cos q* is the independent variable
[this *z* is not the Cartesian coordinate], *Z(z)* is the dependent
variable, and *m* and *n*
are constants. The interval is *-1<z<1*, and the boundary conditions are set
by the system of interest. By comparing Equ (3) with Equ (1) one sees that Legendre's DE
is a special case of Sturm-Liouville problem having *f(x) =
(1-x*^{2}), g(x) = -m^{2} (1-x^{2})^{-1},
l = n(n+1), and *h(x)=1*.

## Solutions

If, as is the case in most physical systems, solutions must be finite at the ends of
the interval, *z = (+1, -1)*, then *n* and *m* are required to be integers. When *m*
*= 0*, the solution is a polynomial, *Z(z) = P*_{n}(z), called the
Legendre polynomial. Otherwise, the solution is called an associated Legendre function: *Z(z)
= P*_{n}^{m}(z). The form of these solutions and some of their
properties are given in convient tables.

### Exercises and problems:

- Use the generating function to determine the first three Legendre Polynomials.
- Use the recurrence formula to determine P
_{6}(x).

Home

Created or up-dated 08/03/99
by R.D. Poshusta