orthogonality of Legendre polynomials
We start from the first order differential equation
(1-x2)dudx+2nxu= 0, | (1) |
where one can separate the variables (http://planetmath.org/SeparationOfVariables) and then get the general solution
u=C(1-x2)n. | (2) |
Differentiating n+1 times the equation (1) it takes the form
(1-x2)dn+2udxn+2-2xdn+1udxn+1+n(n+1)dnudxn= 0 |
or
(1-x2)d2ydx2-2xdydx+n(n+1)y= 0 | (3) |
where
y=dnudxn=Cdndxn(1-x2)n. |
Especially, the particular solution
y=Pn(x):=12nn!dndxn(1-x2)n, | (4) |
which which is the Legendre polynomial of degree n, 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 Pn(x) and to show the orthogonality (http://planetmath.org/OrthogonalPolynomials) of the Legendre polynomials P0(x),P1(x),P2(x),…
0.1 The coefficient of xn
By the binomial theorem,
Pn(x) | =12nn!dndxnn∑j=0(nj)x2(n-j)(-1)j | ||
=12nn!n∑j=0(nj)(2n-2j)(2n-2j-1)⋯(2n-2j-n+1)xn-2j(-1)j. |
From the term with j=0 we get as the coefficient of xn the following:
12nn!(n0)(2n)(2n-1)(2n-2)⋯(2n-n+1)(-1)0=12nn!⋅(2n)!(2n-n)!=(2n)!2n(n!)2 | (5) |
0.2 Orthogonality
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
(6) |
where the integral is gotten from
Thus we infer the recurrence relation
Using this and one easily arrives at
(7) |
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
(8) |
where is the Kronecker delta.
References
- 1 K. Kurki-Suonio: Matemaattiset apuneuvot. Limes r.y., Helsinki (1966).
Title | orthogonality of Legendre polynomials |
---|---|
Canonical name | OrthogonalityOfLegendrePolynomials |
Date of creation | 2013-03-22 18:55:30 |
Last modified on | 2013-03-22 18:55:30 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 14 |
Author | pahio (2872) |
Entry type | Derivation |
Classification | msc 33C45 |
Related topic | OrthogonalPolynomials |
Related topic | OrthogonalityOfChebyshevPolynomials |
Related topic | SubstitutionNotation |