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Appendix B: Evaluation of the Gn integrals

The integrals Gn(a,b) appearing in the expression of the Legendre moments can easily be expressed as sums or differences of the Fermi-like integral
and therefore, the problem is reduced to calculate this kind of integrals.

In order to do this, we expand the denominator in the previous expression (Sack 1990), which must be done in a different way depending on the sign of tex2html_wrap_inline2000, this is:

then, we obtain an infinite sum of integrals that can be calculated analytically using


Let us define the following function
which is well defined for tex2html_wrap_inline2010 and tex2html_wrap_inline2012, we finally arrive to a useful expression for the tex2html_wrap_inline2014 integrals depending on the value of tex2html_wrap_inline1678.

If tex2html_wrap_inline2018

If tex2html_wrap_inline2020

If tex2html_wrap_inline2022

The previous expressions are exact, and we only have to calculate a sum of a finite number of terms (up to k). The accuracy depends exclusively on the evaluation of the tex2html_wrap_inline2026 functions. The fact that these are uniparametric functions allows us to tabulate them in a fine grid at the beginning of the calculation and to obtain enough accuracy without excessive CPU time cost.

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