The Legendre differential equation is the second order ordinary differential equation (ODE) which can be written as:
![{\displaystyle (1-x^{2})d^{2}y/dx^{2}-2xdy/dx+l(l+1)y=0\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/413774def6e1de1c541350dadf28b9ba2e4ca807)
which when rearranged to:
is called Legendre differential equation of order
, where the quantity
is a constant.
![{\displaystyle Ly=0\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c673fb91f941b386e082108869b802b92c83841)
where
is the Legendre operator:
![{\displaystyle L={d \over dx}[(1-x^{2}){d \over dx}]+l(l+1)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d34364604dcad9d6c0c11cfa5f1f279fd5990cdb)
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
|
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.
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Thus
,
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.