5.3 Gaussian Quadrature 125
should correspond to the derivative of the mesh points. Try to convince yourself that the
above expression fulfills this condition.
5.3.5 Other orthogonal polynomials
5.3.5.1 Laguerre polynomials
If we are able to rewrite our integral of Eq. (5.7) with a weight functionW(x) =xαe−xwith
integration limits[ 0 ,∞), we could then use the Laguerre polynomials. The polynomials form
then the basis for the Gauss-Laguerre method which can be applied to integrals of the form
I=∫∞
0
f(x)dx=∫∞
0
xαe−xg(x)dx.These polynomials arise from the solution of the differential equation
(
d^2
dx^2 −
d
dx+λ
x−l(l+ 1 )
x^2)
L(x) = 0 ,wherelis an integerl≥ 0 andλa constant. This equation arises for example from the solution
of the radial Schrödinger equation with a centrally symmetric potential such as the Coulomb
potential. The first few polynomials are
L 0 (x) = 1 ,L 1 (x) = 1 −x,
L 2 (x) = 2 − 4 x+x^2 ,
L 3 (x) = 6 − 18 x+ 9 x^2 −x^3 ,and
L 4 (x) =x^4 − 16 x^3 + 72 x^2 − 96 x+ 24.
They fulfil the orthogonality relation
∫∞
0
e−xLn(x)^2 dx= 1 ,and the recursion relation
(n+ 1 )Ln+ 1 (x) = ( 2 n+ 1 −x)Ln(x)−nLn− 1 (x).5.3.5.2 Hermite polynomials
In a similar way, for an integral which goes like
I=∫∞
−∞
f(x)dx=∫∞
−∞
e−x
2
g(x)dx.we could use the Hermite polynomials in order to extract weights and mesh points. The
Hermite polynomials are the solutions of the following differential equation
d^2 H(x)
dx^2
− 2 xdH(x)
dx
+ (λ− 1 )H(x) = 0.