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* Page found: Das ideale Fermigas (eq math.2555.86)

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TeX (original user input): 

\begin{align}

& {{F}_{s}}\left( \eta  \right)=\frac{1}{\Gamma \left( s+1 \right)}\int_{0}^{\infty }{{}}dy\frac{{{y}^{s}}}{{{e}^{y-\eta }}+1} \\

& =\frac{1}{\Gamma \left( s+1 \right)}\int_{0}^{\infty }{{}}dy{{y}^{s}}\frac{\xi {{e}^{-y}}}{1+\xi {{e}^{-y}}}\approx \frac{1}{\Gamma \left( s+1 \right)}\left[ \xi \int_{0}^{\infty }{{}}dy{{y}^{s}}{{e}^{-y}}-{{\xi }^{2}}\int_{0}^{\infty }{{}}dy{{y}^{s}}{{e}^{-2y}}+.... \right] \\

& \int_{0}^{\infty }{{}}dy{{y}^{s}}{{e}^{-y}}=\Gamma \left( s+1 \right) \\

& \int_{0}^{\infty }{{}}dy{{y}^{s}}{{e}^{-2y}}=\frac{1}{{{2}^{s+1}}}\int_{0}^{\infty }{{}}dz{{z}^{s}}{{e}^{-z}}=\frac{1}{{{2}^{s+1}}}\Gamma \left( s+1 \right) \\

& \Rightarrow {{F}_{s}}\left( \eta  \right)\approx \left[ \xi -{{\xi }^{2}}\frac{1}{{{2}^{s+1}}}+.... \right]\approx \left[ \xi -{{\xi }^{2}}\frac{1}{{{2}^{s+1}}} \right]={{e}^{\frac{\mu }{kT}}}\left[ 1-{{e}^{\frac{\mu }{kT}}}\frac{1}{{{2}^{s+1}}} \right] \\

\end{align}

TeX (checked): 

{\begin{aligned}&{{F}_{s}}\left(\eta \right)={\frac {1}{\Gamma \left(s+1\right)}}\int _{0}^{\infty }{}dy{\frac {{y}^{s}}{{{e}^{y-\eta }}+1}}\\&={\frac {1}{\Gamma \left(s+1\right)}}\int _{0}^{\infty }{}dy{{y}^{s}}{\frac {\xi {{e}^{-y}}}{1+\xi {{e}^{-y}}}}\approx {\frac {1}{\Gamma \left(s+1\right)}}\left[\xi \int _{0}^{\infty }{}dy{{y}^{s}}{{e}^{-y}}-{{\xi }^{2}}\int _{0}^{\infty }{}dy{{y}^{s}}{{e}^{-2y}}+....\right]\\&\int _{0}^{\infty }{}dy{{y}^{s}}{{e}^{-y}}=\Gamma \left(s+1\right)\\&\int _{0}^{\infty }{}dy{{y}^{s}}{{e}^{-2y}}={\frac {1}{{2}^{s+1}}}\int _{0}^{\infty }{}dz{{z}^{s}}{{e}^{-z}}={\frac {1}{{2}^{s+1}}}\Gamma \left(s+1\right)\\&\Rightarrow {{F}_{s}}\left(\eta \right)\approx \left[\xi -{{\xi }^{2}}{\frac {1}{{2}^{s+1}}}+....\right]\approx \left[\xi -{{\xi }^{2}}{\frac {1}{{2}^{s+1}}}\right]={{e}^{\frac {\mu }{kT}}}\left[1-{{e}^{\frac {\mu }{kT}}}{\frac {1}{{2}^{s+1}}}\right]\\\end{aligned}}

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F s ⁢ ( η ) = 1 Γ ⁢ ( s + 1 ) ⁢ ∫ 0 ∞ 𝑑 y ⁢ y s e y - η + 1 = 1 Γ ⁢ ( s + 1 ) ∫ 0 ∞ d y y s ξ ⁢ e - y 1 + ξ ⁢ e - y ≈ 1 Γ ⁢ ( s + 1 ) [ ξ ∫ 0 ∞ d y y s e - y - ξ 2 ∫ 0 ∞ d y y s e - 2 ⁢ y + … . ] ∫ 0 ∞ 𝑑 y ⁢ y s ⁢ e - y = Γ ⁢ ( s + 1 ) ∫ 0 ∞ 𝑑 y ⁢ y s ⁢ e - 2 ⁢ y = 1 2 s + 1 ⁢ ∫ 0 ∞ 𝑑 z ⁢ z s ⁢ e - z = 1 2 s + 1 ⁢ Γ ⁢ ( s + 1 ) ⇒ F s ( η ) ≈ [ ξ - ξ 2 1 2 s + 1 + … . ] ≈ [ ξ - ξ 2 1 2 s + 1 ] = e μ k ⁢ T [ 1 - e μ k ⁢ T 1 2 s + 1 ] absent subscript 𝐹 𝑠 𝜂 1 Γ 𝑠 1 superscript subscript 0 differential-d 𝑦 superscript 𝑦 𝑠 superscript 𝑒 𝑦 𝜂 1 absent fragments 1 Γ 𝑠 1 superscript subscript 0 d y superscript 𝑦 𝑠 𝜉 superscript 𝑒 𝑦 1 𝜉 superscript 𝑒 𝑦 1 Γ 𝑠 1 fragments [ ξ superscript subscript 0 d y superscript 𝑦 𝑠 superscript 𝑒 𝑦 superscript 𝜉 2 superscript subscript 0 d y superscript 𝑦 𝑠 superscript 𝑒 2 𝑦 … . ] absent superscript subscript 0 differential-d 𝑦 superscript 𝑦 𝑠 superscript 𝑒 𝑦 Γ 𝑠 1 absent superscript subscript 0 differential-d 𝑦 superscript 𝑦 𝑠 superscript 𝑒 2 𝑦 1 superscript 2 𝑠 1 superscript subscript 0 differential-d 𝑧 superscript 𝑧 𝑠 superscript 𝑒 𝑧 1 superscript 2 𝑠 1 Γ 𝑠 1 absent fragments ⇒ subscript 𝐹 𝑠 fragments ( η ) fragments [ ξ superscript 𝜉 2 1 superscript 2 𝑠 1 … . ] fragments [ ξ superscript 𝜉 2 1 superscript 2 𝑠 1 ] superscript 𝑒 𝜇 𝑘 𝑇 fragments [ 1 superscript 𝑒 𝜇 𝑘 𝑇 1 superscript 2 𝑠 1 ] {\displaystyle{\displaystyle\begin{aligned} \par&\displaystyle{{F}_{s}}\left(% \eta\right)=\frac{1}{\Gamma\left(s+1\right)}\int_{0}^{\infty}{{}}dy\frac{{{y}^% {s}}}{{{e}^{y-\eta}}+1}\\ \par&\displaystyle=\frac{1}{\Gamma\left(s+1\right)}\int_{0}^{\infty}{{}}dy{{y}% ^{s}}\frac{\xi{{e}^{-y}}}{1+\xi{{e}^{-y}}}\approx\frac{1}{\Gamma\left(s+1% \right)}\left[\xi\int_{0}^{\infty}{{}}dy{{y}^{s}}{{e}^{-y}}-{{\xi}^{2}}\int_{0% }^{\infty}{{}}dy{{y}^{s}}{{e}^{-2y}}+....\right]\\ \par&\displaystyle\int_{0}^{\infty}{{}}dy{{y}^{s}}{{e}^{-y}}=\Gamma\left(s+1% \right)\\ \par&\displaystyle\int_{0}^{\infty}{{}}dy{{y}^{s}}{{e}^{-2y}}=\frac{1}{{{2}^{s% +1}}}\int_{0}^{\infty}{{}}dz{{z}^{s}}{{e}^{-z}}=\frac{1}{{{2}^{s+1}}}\Gamma% \left(s+1\right)\\ \par&\displaystyle\Rightarrow{{F}_{s}}\left(\eta\right)\approx\left[\xi-{{\xi}% ^{2}}\frac{1}{{{2}^{s+1}}}+....\right]\approx\left[\xi-{{\xi}^{2}}\frac{1}{{{2% }^{s+1}}}\right]={{e}^{\frac{\mu}{kT}}}\left[1-{{e}^{\frac{\mu}{kT}}}\frac{1}{% {{2}^{s+1}}}\right]\\ \par\end{aligned}}}

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Fs(η)=1Γ(s+1)∫0∞dyysey−η+1=1Γ(s+1)∫0∞dyysξe−y1+ξe−y≈1Γ(s+1)[ξ∫0∞dyyse−y−ξ2∫0∞dyyse−2y+....]∫0∞dyyse−y=Γ(s+1)∫0∞dyyse−2y=12s+1∫0∞dzzse−z=12s+1Γ(s+1)⇒Fs(η)≈[ξ−ξ212s+1+....]≈[ξ−ξ212s+1]=eμkT[1−eμkT12s+1]

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