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Legendre differential equation

The Legendre differential equation is the second order ordinary differential equation (ODE) which can be written as:

which can be written as:

is called Legendre differential equation of order l, in which the quantity l is a constant.

where is the Legendre operator:

In principle, l 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


.


Thus


,

and in general,

.

This series converges when

.


Therefore the series solution has to be cut by choosing:


.


The series cut in specific integers l and l+1 produce polynomials called Legendre polynomials.