Gauss-Laguerre on Plank's Law

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 [1].

I think we can approximate the integral as

Beyer (1987) gives a table of abscissas and weights up to n=6 [2].

Substituting values to approximate the integral we get

=6.494316315

This is very close to the value computed using the composite Simpson quadrature rule.

[2] Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 463, 1987..