Mathematical integral
Inmathematics , thecomplete Fermi–Dirac integral , named afterEnrico Fermi andPaul Dirac , for an indexj is defined by
F j ( x ) = 1 Γ ( j + 1 ) ∫ 0 ∞ t j e t − x + 1 d t , ( j > − 1 ) {\displaystyle F_{j}(x)={\frac {1}{\Gamma (j+1)}}\int _{0}^{\infty }{\frac {t^{j}}{e^{t-x}+1}}\,dt,\qquad (j>-1)} This equals
− Li j + 1 ( − e x ) , {\displaystyle -\operatorname {Li} _{j+1}(-e^{x}),} whereLi s ( z ) {\displaystyle \operatorname {Li} _{s}(z)} polylogarithm .
Its derivative is
d F j ( x ) d x = F j − 1 ( x ) , {\displaystyle {\frac {dF_{j}(x)}{dx}}=F_{j-1}(x),} and this derivative relationship is used to define the Fermi-Dirac integral for nonpositive indicesj . Differing notation forF j {\displaystyle F_{j}} 1 / Γ ( j + 1 ) {\displaystyle 1/\Gamma (j+1)} NIST DLMF .
The closed form of the function exists forj = 0:
F 0 ( x ) = ln ( 1 + exp ( x ) ) . {\displaystyle F_{0}(x)=\ln(1+\exp(x)).} Forx = 0 , the result reduces to
F j ( 0 ) = η ( j + 1 ) , {\displaystyle F_{j}(0)=\eta (j+1),}
whereη {\displaystyle \eta } Dirichlet eta function .
Gradshteyn, Izrail Solomonovich ;Ryzhik, Iosif Moiseevich ;Geronimus, Yuri Veniaminovich ;Tseytlin, Michail Yulyevich ; Jeffrey, Alan (2015) [October 2014]. "3.411.3.". In Zwillinger, Daniel;Moll, Victor Hugo (eds.).Table of Integrals, Series, and Products Academic Press, Inc. p. 355.ISBN 978-0-12-384933-5 LCCN 2014010276 .ISBN 978-0-12-384933-5 .R.B.Dingle (1957).Fermi-Dirac Integrals . Appl.Sci.Res. B6. pp. 225– 239.
