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Looking up some mathematical results on ellipsoids, it seems that you need to define some elliptic integrals which don't have an analytic solution. But I also found a PDF from some other semiconductor course which outlines this exact problem, and then gives a simple result that the density of states is proportional to sqrt(m*1 m*2 m*3) for the case where you have a general ellipsoid. Am I missing some major simplifying trick in the calculation?

Thanks.