Roy Vincent L. Canseco, MSEE Jan 2014
We want to approximate the integral of Plank's theory on black-body radiation using Gauss-Laguerre quadrature.
Plank’s law for black-body radiation takes the form
Plotting the integrand
The Gauss-Laguerre Formula uses Laguerre polynomials to treat the integration interval of as shown below .
I think we can approximate the integral as
Beyer (1987) gives a table of abscissas and weights up to n=6 .
Substituting values to approximate the integral we get
This is very close to the value computed using the composite Simpson quadrature rule.
 Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 463, 1987..