Evaluation of some double FD integrals
Evaluation of double Fermi-Dirac integrals. e.g., ...
∫-∞y F-1/2(x)
F-1/2(x)
dx
or more
generally ...
[Γ(m)]-1
∫-∞y (y-x)m-1 Fj(x)
Fk(x) dx =
Gm,jk(y) , where m>0.
Such integrals
arise in calculations of pairwise fermion interactions in condensed
matter systems, e.g., when calculating the electron EM exchange
energy (see, e.g., the 'specfunc.ElectronDegPres' example
and § 80 in Landau
and Lifshitz, Statistical Physics Part 1 (3rd edition),
1980).
The values of y and m can be
adjusted via the integrand parameters during model runtime. The
values of j and k, however, must be chosen before creating
(compiling) the integrand (see 'DoubleFDIntegrand').
Note that here
-ymax is chosen instead of -∞ since for most such
integrals, very little contribution arises from the x<<-1
region.
For values of m<1, care must
be taken with the choice of integration routine as a singularity at
x=y will appear in the integrand.
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