Über das asymptotische Verhalten des Fermi-Dirac-Integrals. (On the asymptotical behaviour of the Fermi-Dirac integral)
type: Item
instance of: scholarly article
title: Über das asymptotische Verhalten des Fermi-Dirac-Integrals. (On the asymptotical behaviour of the Fermi-Dirac integral)
zbMATH Open document ID: 0638.41029
DOI: 10.4171/ZAA/261
author: H.-J. Schell
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1987-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 41A60
zbMATH DE Number: 4040493
zbMATH Keywords: Fermi-Dirac integral
zbMATH Keywords: incomplete Gamma function
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4040493