orthogonality of Laguerre polynomials
The polynomials (1) themselves are not orthogonal to each other, but the expressions () are orthogonal (http://planetmath.org/OrthogonalPolynomials) on the interval from 0 to , i.e. the polynomials are orthogonal with respect to the weighting function on that interval, as is seen in the following.
Let be another nonnegative integer. We integrate by parts (http://planetmath.org/IntegrationByParts) times in
When , this yields
and for it gives
The result (2) implies, because is a polynomial of degree , that
Thus the orthogonality has been shown. Therefore, since the leading term of is , we infer by (3) and (4) that
so that the expressions form a system of orthonormal polynomials.
- 1 H. Eyring, J. Walter, G. Kimball: Quantum chemistry. Eight printing. Wiley & Sons, New York (1958).
|Title||orthogonality of Laguerre polynomials|
|Date of creation||2013-03-22 19:05:50|
|Last modified on||2013-03-22 19:05:50|
|Last modified by||pahio (2872)|