For an ideal gas of fermions, we had shown that the problem of determining the equation of state was one of solving two equations
where the second of these must be solved for 
in terms of 
and substituted into the first to obtain P as a function of .
As we did in the Boltzmann case, let us consider the thermodynamic limit
so that the spacing between energy levels becomes
small. Then the sums can be replaced by integrals over the continuous
variable 
. For the pressure, this replacement give rise to
Change variables to
Then,
The remaining integral can be evaluated by expanding the log in a power series and integrating the series term by term:
By the same technique, the average particle number 
can be
shown to be equal to
Multipling both over these equations on both sides by 
gives
Although exact solution of these equations analytically is intractable, we will consider their solutions in two interesting limits: The high temperature, low density limit and its counterpart, the low temperature, high density limit.