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* Page found: Das ideale Fermigas (eq math.2549.49)

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\begin{align}

& \Gamma \left( s+1 \right){{F}_{s}}\left( \eta  \right)=\frac{1}{s+1}\int_{-\eta }^{\infty }{{}}dx{{\left( x+\eta  \right)}^{s+1}}\frac{{{e}^{x}}}{{{\left( {{e}^{x}}+1 \right)}^{2}}}\approx \frac{1}{s+1}\int_{-\infty }^{\infty }{{}}dx{{\left( x+\eta  \right)}^{s+1}}\frac{{{e}^{x}}}{{{\left( {{e}^{x}}+1 \right)}^{2}}}+O\left( {{e}^{-\eta }} \right) \\

& \approx \frac{1}{s+1}\int_{-\infty }^{\infty }{{}}dx{{\left( \eta  \right)}^{s+1}}\frac{{{e}^{x}}}{{{\left( {{e}^{x}}+1 \right)}^{2}}}+\int_{-\infty }^{\infty }{{}}dx{{\left( \eta  \right)}^{s}}x\frac{{{e}^{x}}}{{{\left( {{e}^{x}}+1 \right)}^{2}}}+\frac{s}{2}\int_{-\infty }^{\infty }{{}}dx{{\left( \eta  \right)}^{s-1}}{{x}^{2}}\frac{{{e}^{x}}}{{{\left( {{e}^{x}}+1 \right)}^{2}}} \\

& =\frac{{{\left( \eta  \right)}^{s+1}}}{s+1}\int_{-\infty }^{\infty }{{}}dx\frac{{{e}^{x}}}{{{\left( {{e}^{x}}+1 \right)}^{2}}}+{{\left( \eta  \right)}^{s}}\int_{-\infty }^{\infty }{{}}dx\frac{x{{e}^{x}}}{{{\left( {{e}^{x}}+1 \right)}^{2}}}+\frac{s}{2}{{\left( \eta  \right)}^{s-1}}\int_{-\infty }^{\infty }{{}}dx{{x}^{2}}\frac{{{e}^{x}}}{{{\left( {{e}^{x}}+1 \right)}^{2}}} \\

\end{align}

TeX (checked): 

{\begin{aligned}&\Gamma \left(s+1\right){{F}_{s}}\left(\eta \right)={\frac {1}{s+1}}\int _{-\eta }^{\infty }{}dx{{\left(x+\eta \right)}^{s+1}}{\frac {{e}^{x}}{{\left({{e}^{x}}+1\right)}^{2}}}\approx {\frac {1}{s+1}}\int _{-\infty }^{\infty }{}dx{{\left(x+\eta \right)}^{s+1}}{\frac {{e}^{x}}{{\left({{e}^{x}}+1\right)}^{2}}}+O\left({{e}^{-\eta }}\right)\\&\approx {\frac {1}{s+1}}\int _{-\infty }^{\infty }{}dx{{\left(\eta \right)}^{s+1}}{\frac {{e}^{x}}{{\left({{e}^{x}}+1\right)}^{2}}}+\int _{-\infty }^{\infty }{}dx{{\left(\eta \right)}^{s}}x{\frac {{e}^{x}}{{\left({{e}^{x}}+1\right)}^{2}}}+{\frac {s}{2}}\int _{-\infty }^{\infty }{}dx{{\left(\eta \right)}^{s-1}}{{x}^{2}}{\frac {{e}^{x}}{{\left({{e}^{x}}+1\right)}^{2}}}\\&={\frac {{\left(\eta \right)}^{s+1}}{s+1}}\int _{-\infty }^{\infty }{}dx{\frac {{e}^{x}}{{\left({{e}^{x}}+1\right)}^{2}}}+{{\left(\eta \right)}^{s}}\int _{-\infty }^{\infty }{}dx{\frac {x{{e}^{x}}}{{\left({{e}^{x}}+1\right)}^{2}}}+{\frac {s}{2}}{{\left(\eta \right)}^{s-1}}\int _{-\infty }^{\infty }{}dx{{x}^{2}}{\frac {{e}^{x}}{{\left({{e}^{x}}+1\right)}^{2}}}\\\end{aligned}}

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Γ ⁢ ( s + 1 ) ⁢ F s ⁢ ( η ) = 1 s + 1 ⁢ ∫ - η ∞ 𝑑 x ⁢ ( x + η ) s + 1 ⁢ e x ( e x + 1 ) 2 ≈ 1 s + 1 ⁢ ∫ - ∞ ∞ 𝑑 x ⁢ ( x + η ) s + 1 ⁢ e x ( e x + 1 ) 2 + O ⁢ ( e - η ) ≈ 1 s + 1 ⁢ ∫ - ∞ ∞ 𝑑 x ⁢ ( η ) s + 1 ⁢ e x ( e x + 1 ) 2 + ∫ - ∞ ∞ 𝑑 x ⁢ ( η ) s ⁢ x ⁢ e x ( e x + 1 ) 2 + s 2 ⁢ ∫ - ∞ ∞ 𝑑 x ⁢ ( η ) s - 1 ⁢ x 2 ⁢ e x ( e x + 1 ) 2 = ( η ) s + 1 s + 1 ⁢ ∫ - ∞ ∞ 𝑑 x ⁢ e x ( e x + 1 ) 2 + ( η ) s ⁢ ∫ - ∞ ∞ 𝑑 x ⁢ x ⁢ e x ( e x + 1 ) 2 + s 2 ⁢ ( η ) s - 1 ⁢ ∫ - ∞ ∞ 𝑑 x ⁢ x 2 ⁢ e x ( e x + 1 ) 2 absent Γ 𝑠 1 subscript 𝐹 𝑠 𝜂 1 𝑠 1 superscript subscript 𝜂 differential-d 𝑥 superscript 𝑥 𝜂 𝑠 1 superscript 𝑒 𝑥 superscript superscript 𝑒 𝑥 1 2 1 𝑠 1 superscript subscript differential-d 𝑥 superscript 𝑥 𝜂 𝑠 1 superscript 𝑒 𝑥 superscript superscript 𝑒 𝑥 1 2 𝑂 superscript 𝑒 𝜂 absent absent 1 𝑠 1 superscript subscript differential-d 𝑥 superscript 𝜂 𝑠 1 superscript 𝑒 𝑥 superscript superscript 𝑒 𝑥 1 2 superscript subscript differential-d 𝑥 superscript 𝜂 𝑠 𝑥 superscript 𝑒 𝑥 superscript superscript 𝑒 𝑥 1 2 𝑠 2 superscript subscript differential-d 𝑥 superscript 𝜂 𝑠 1 superscript 𝑥 2 superscript 𝑒 𝑥 superscript superscript 𝑒 𝑥 1 2 absent absent superscript 𝜂 𝑠 1 𝑠 1 superscript subscript differential-d 𝑥 superscript 𝑒 𝑥 superscript superscript 𝑒 𝑥 1 2 superscript 𝜂 𝑠 superscript subscript differential-d 𝑥 𝑥 superscript 𝑒 𝑥 superscript superscript 𝑒 𝑥 1 2 𝑠 2 superscript 𝜂 𝑠 1 superscript subscript differential-d 𝑥 superscript 𝑥 2 superscript 𝑒 𝑥 superscript superscript 𝑒 𝑥 1 2 {\displaystyle{\displaystyle\begin{aligned} \par&\displaystyle\Gamma\left(s+1% \right){{F}_{s}}\left(\eta\right)=\frac{1}{s+1}\int_{-\eta}^{\infty}{{}}dx{{% \left(x+\eta\right)}^{s+1}}\frac{{{e}^{x}}}{{{\left({{e}^{x}}+1\right)}^{2}}}% \approx\frac{1}{s+1}\int_{-\infty}^{\infty}{{}}dx{{\left(x+\eta\right)}^{s+1}}% \frac{{{e}^{x}}}{{{\left({{e}^{x}}+1\right)}^{2}}}+O\left({{e}^{-\eta}}\right)% \\ \par&\displaystyle\approx\frac{1}{s+1}\int_{-\infty}^{\infty}{{}}dx{{\left(% \eta\right)}^{s+1}}\frac{{{e}^{x}}}{{{\left({{e}^{x}}+1\right)}^{2}}}+\int_{-% \infty}^{\infty}{{}}dx{{\left(\eta\right)}^{s}}x\frac{{{e}^{x}}}{{{\left({{e}^% {x}}+1\right)}^{2}}}+\frac{s}{2}\int_{-\infty}^{\infty}{{}}dx{{\left(\eta% \right)}^{s-1}}{{x}^{2}}\frac{{{e}^{x}}}{{{\left({{e}^{x}}+1\right)}^{2}}}\\ \par&\displaystyle=\frac{{{\left(\eta\right)}^{s+1}}}{s+1}\int_{-\infty}^{% \infty}{{}}dx\frac{{{e}^{x}}}{{{\left({{e}^{x}}+1\right)}^{2}}}+{{\left(\eta% \right)}^{s}}\int_{-\infty}^{\infty}{{}}dx\frac{x{{e}^{x}}}{{{\left({{e}^{x}}+% 1\right)}^{2}}}+\frac{s}{2}{{\left(\eta\right)}^{s-1}}\int_{-\infty}^{\infty}{% {}}dx{{x}^{2}}\frac{{{e}^{x}}}{{{\left({{e}^{x}}+1\right)}^{2}}}\\ \par\end{aligned}}}

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Γ(s+1)Fs(η)=1s+1∫−η∞dx(x+η)s+1ex(ex+1)2≈1s+1∫−∞∞dx(x+η)s+1ex(ex+1)2+O(e−η)≈1s+1∫−∞∞dx(η)s+1ex(ex+1)2+∫−∞∞dx(η)sxex(ex+1)2+s2∫−∞∞dx(η)s−1x2ex(ex+1)2=(η)s+1s+1∫−∞∞dxex(ex+1)2+(η)s∫−∞∞dxxex(ex+1)2+s2(η)s−1∫−∞∞dxx2ex(ex+1)2

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