Evaluation of Fermi—Dirac Integral

Abstract

The Fermi-Dirac integral is defined by

$$ {F_\mu }(x): = \int_0^x {\frac{{{x^\mu }dx}}{{1 + {e^{x - x}}}}} (\mu > - 1). $$

Some exact expressions for F _μ are well known. Basing on them one obtains some new approximate expressions. Three cases are distinguished:

Case a $\mu = 1,2,...,x \geqslant 0$. The non-polynomial part of F _μ (z) is expanded, after a suitable variable transformation, into Chebyshev series (Section 2).

Case b $\mu = - \frac{1}{2},\frac{1}{2},...,x \geqslant 0$is sufficiently small). F _μ (z) is expanded in powers of $x = 1 - {(1 + {e^x})^{\frac{1}{2}}}$. Then Padé approximation is used (Section 3).

Case c $\mu = - \frac{1}{2},\frac{1}{2},...,x \geqslant {u^2}$, with a sufficiently large u). Fμ(z) is expanded, at first, into a series containing the fonctions Erfi and Eric and, after that, into Chebyshev series with the variable $u/\sqrt z$(Section 4).

These methods seem to be more general and, from some points of view, better than the earlier ones.