Über das asymptotische Verhalten des Fermi-Dirac-Integrals. (On the asymptotical behaviour of the Fermi-Dirac integral) (Q1099349)

Asymptotic expansions of the Fermi-Dirac integral \[ F_{p- 1}(x)=(\int^{\infty}_{0}t^{p-1}/(1+\exp (t-x))dt)/\Gamma (p), \] \(p>0\) real, x real, are derived when both of the parameters x, p tend to infinity, namely for \(p\sim x\) and for \(p=ax\), \(0<a<2\). A new asymptotic approximation is given holding in case that \(p\to \infty\), \(\sqrt{p}=o(p- 2x)\). Furthermore the optimal truncation term of the well-known asymptotic expansion for \(x\to \infty\), p fixed, is discussed. Finally an asymptotic relation between \(F_{p-1}(x)\) and the incomplete Gamma function is pointed out.