Fermi integrals

• To: mathgroup at yoda.physics.unc.edu
• Subject: Fermi integrals
• From: "Tim C" <tc at br4130mail.nrel.gov>
• Date: 24 Nov 1993 13:43:04 -0700

```I'd appreciate some help with the evaluation of the Fermi integrals, which are
of the form:
a[ef_, te_] := NIntegrate[e^n (Exp[(e - ef)/(k te)] + 1)^-1, {e, 0, Infinity}].

In this expression, ef is the quantity to be determined, k and  te are
constants, which I can input.  n is an odd integer multiple of 1/2.  Take n to
be unity to start with.  Essentially, I know the numerical value of the
integral and to need to know the value of ef that will balance the equation. I
have achieved this be tabulating the integral for a range of values of ef that
I know is sufficiently large to cover all practical situations.  I then set up
an interpolation function and get ef to an easily acceptable accuracy.  On my
Macintosh II-fx, this takes about 65 seconds to do!  I really need to reduce
this be a substantial factor since the expression may be called quite a few
times in the overall program.

Even if I reduce the upper limit of the integral to unity, which I would rather
not do but which is adequate for most practical cases, it still takes far too
long.  Can anyone suggest any ideas on how I might speed things up?  I'd be
delighted to get some new approaches.  Thanks in advance.

Tim Coutts,
National Renewable Energy Laboratory,
Golden,