orthogonality of Legendre polynomials
where one can separate the variables (http://planetmath.org/SeparationOfVariables) and then get the general solution
Differentiating times the equation (1) it takes the form
Especially, the particular solution
which which is the Legendre polynomial of degree , has been seen to satisfy the Legendre’s differential equation (3).
The equality (4) is Rodrigues formula (http://planetmath.org/RodriguesFormula). We use it to find the leading coefficient of and to show the orthogonality (http://planetmath.org/OrthogonalPolynomials) of the Legendre polynomials
0.1 The coefficient of
By the binomial theorem,
From the term with we get as the coefficient of the following:
Let be any polynomial of degree . Integrating by parts (http://planetmath.org/IntegrationByParts) times we obtain
since are zeros of the derivatives .
If, on the other hand, , the calculation gives firstly
where the integral is gotten from
Thus we infer the recurrence relation
Using this and one easily arrives at
If also is a Legendre polynomial , we can in (6) by (5) put
and taking into account (7), too, (6) reads
Our results imply the orthonormality (http://planetmath.org/Orthonormal) condition
where is the Kronecker delta.
- 1 K. Kurki-Suonio: Matemaattiset apuneuvot. Limes r.y., Helsinki (1966).
|Title||orthogonality of Legendre polynomials|
|Date of creation||2013-03-22 18:55:30|
|Last modified on||2013-03-22 18:55:30|
|Last modified by||pahio (2872)|