Legendre differential equation
The Legendre differential equation is the second order ordinary differential equation (ODE) which can be written as:
which when rearranged to:
- is called Legendre differential equation of order , where the quantity is a constant.
where is the Legendre operator:
In principle, can be any number, but it is usually an integer.
We use the Frobenius method to solve the equation in the region . We start by setting the parameter p in Frobenius method zero.
Substituting these terms into the original equation, one obtains
and in general,
This series converges when
Therefore the series solution has to be cut by choosing:
The series cut in specific integers and produce polynomials called Legendre polynomials.