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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].


n

Xi

Wi

2

0.585786

0.853553


3.41421

0.146447

3

0.415775

0.711093


2.29428

0.278518


6.28995

0.0103893

4

0.322548

0.603154


1.74576

0.357419


4.53662

0.0388879


9.39507

0.000539295

5

0.26356

0.521756


1.4134

0.398667


3.59643

0.0759424


7.08581

0.00361176


12.6408

0.00002337





Substituting values to approximate the integral we get


=6.494316315


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



[1] http://www.efunda.com/math/num_integration/num_int_gauss.cfm

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

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