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Sorry it took me so long to respond, life circumstances intervened.

First, sorry if the question wasn't clear enough. This is the first time that I have posted here so I am unfamiliar with the level of expertise in the community, which makes it difficult to write a specific, targeted question.

My original question had two parts:

1. Can I use Sage to run the GSL algorithm to evaluate the Fermi-Dirac integral of half-order? I am only interested in the solution of this specific integral, not the solution of any other Fermi-Dirac integral. Other orders have analytical solutions, but the Fermi-Dirac integral of half order does not; it must be done numerically. Here is one reference asserting the same claim (click on it, press ctrl+f and type in numerical to get to the specific sentence): nanohub.org

2. If I cannot use the GSL function, what other options are there?

The first has gone unanswered. The second has been soundly answered and I am grateful for your response! Barring an affirmative answer to the first question, I will use the numerical_integral command to evaluate this integral.

I am, however, troubled by your comments in general. I like SAGE because it is a free and open-source alternative to very expensive software packages like MM. Any work that I do in MM I lose as soon as I don't work somewhere that maintains a MM license, which is almost anywhere other than academia. Suggesting that I use MM seems to go against the stated purpose of SAGE and it's creator's original intent. I understand that in certain circumstances it is a more capable tool. That's fine, but SAGE includes at least 3 packages dedicated to numerical computation (octave, numpy, and scilab). I thought I would at least get a suggestion about how to use one of those tools as an alternative to the GSL algorithm before running to MM with my tail between my legs.

Further, your suggestion that MM "knows" how to perform this integral analytically and does so is an egregious error. The Fermi-Dirac integral of negative half order (used in your example) must also be done numerically (quick google search). Even the answer given by MM, a number and not an expression, suggests that it is not employing a symbolic solution, but is indeed switching to a numerical method. It's equally possible that MM is programmed to recognize Fermi-Dirac integrals and respond accordingly. It should be noted that I found no evidence in the MM documentation to suggest that either one of our answers is correct. This is the core problem of the MM black box, and another reason why SAGE is valuable. In this case, the Maxima engine in SAGE is outperforming MM by providing a more accurate and predictable (although useless) outcome. The only thing the MM result confirms is that the numerical_integral method in SAGE provides the same answer. This suggests that MM is correct because it can be compared against a verified and understood method contained in SAGE. Note that this is the same conclusion (that the answer is correct) but applied in reverse because the solution which has been tested and validated by a community of researchers is more valuable to me.