In the Sturm-Liouville Boundary Value Problem, there is an important special case called Legendre's Differential Equation which arises in numerous problems, especially in those exhibiting spherical symmetry. Legendre's Differential Equation is defined as:
where is a real number. The solutions of this equation are called Legendre Functions of degree .
When is a non-negative integer, i.e., , the Legendre Functions are often referred to as Legendre Polynomials .
Since Legendre's differential equation is a second order ordinary differential equation, two sets of functions are needed to form the general solution. Legendre Polynomials of the second kind are then introduced. The general solution of a non-negative integer degree Legendre's Differential Equation can hence be expressed as:
However, is divergent at . Therefore, the associated coefficient is forced to be zero to obtain a physically meaningful result when there are no sources or sinks at the boundary points .
Rodrigues' Formula: The Legendre Polynomials can be expressed by Rodrigues' formula
Generating Function: The generating function of a Legendre Polynomial is
Orthogonality: Legendre Polynomials , , form a complete orthogonal set on the interval . It can be shown that
By using this orthogonality, a piecewise continuous function in can be expressed in terms of Legendre Polynomials:
This orthogonal series expansion is also known as a Fourier-Legendre Series expansion or a Generalized Fourier Series expansion.
Even/Odd Functions: Whether a Legendre Polynomial is an even or odd function depends on its degree .
Based on ,
is an even function when is even.
is an odd function when is odd.
In addition, from,
is an even function when is odd.
is an odd function when is even.
Recurrence Relation: A Legendre Polynomial at one point can be expressed by neighboring Legendre Polynomials at the same point.