Complete Fermi–Dirac integral

In statistical physics, the Fermi -Dirac integral is defined (after Enrico Fermi and Paul Dirac ) with index j as

Where the gamma function. The lower limit of the integral of the function given as an argument

Then one speaks of the incomplete Fermi -Dirac integral.

Application for F1 / 2

The function occurs, inter alia, in solid state physics in connection with the residence distribution of electrons in the crystal lattice. There is often the integral to be calculated (see density ). Substituiere the second equal sign as well, so that:

Approximation for F1 / 2

The integral can be solved for different ranges of values ​​of x approximation:

The relative error of this approximation solution is at most 3% ( maximum deviation at and at ). For large distance from the origin can be approximated by two functions: