|
Johnson and Branstetter favor Laguerre-Gauss quadrature over the usual series summation2 for the fast, precise evaluation of integrals of the Planck function. But the only disadvantage of the usual series, which is an expansion in powers of exp (-x), is that it does not converge very rapidly for small x. This problem is easily solved through the use, for small x, of a more rapidly converging expansion in powers of x. The two series together provide more precision than the Laguerre-Gauss quadrature for a comparable number of terms, and are fast and simple to use. This note summarizes the expressions needed to obtain any required accuracy with the two-series approach. Expressions are given which bound the fractional residual errors when the series are truncated; and to show the precision attainable, these bounds are plotted as functions of x for 5 and 10 terms of each series.
|
