If g( ?) is a monotonically increasing function of ? whose value is zero when ? is zero, and f( ?) is the Fermi function defined by

then the Fermi integral is defined by

or

Equation (A.5.3) is obtained from (A.5.2) by an integration by parts, and F( ?) is defined by

Since g( ?) is given, F( ?) is a known function. Examples of Fermi integrals are those with g( ?) given by ?? and ? ^3/2, which are used in getting the Fermi energy and energy as a function of temperature for a gas of free electrons.

From the form of the Fermi function, the derivative ? f/ ?? is practically zero for all energies except in the vicinity of the Fermi energy, near ? = ?. This means that a rapidly converging series can be obtained by expanding F( ?) in a Taylor series about ?. Start with the Taylor expansion:

F _r( ?) being the rth derivative of F( ?) evaluated at ? = ?. That is,

Equation (A.5.3) can now be written as

I _r being defined by

Now let us work on this integral. From the definition of the Fermi function, the derivative is

Define a variable z by

so that (A.5.9) becomes

Using this in (A.5.8) and putting the result in...