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Display information for equation id:math.2555.43 on revision:2555

* Page found: Das ideale Fermigas (eq math.2555.43)

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TeX (original user input): 

\begin{align}

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

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

& \frac{1}{s+1}\left. \left[ \left( {{y}^{s+1}} \right)\frac{1}{\left( {{e}^{y-\eta }}+1 \right)} \right] \right|_{0}^{\infty }=0 \\

\end{align}

TeX (checked): 

{\begin{aligned}&\Gamma \left(s+1\right){{F}_{s}}\left(\eta \right):=\int _{0}^{\infty }{}dy{\frac {{y}^{s}}{\left({{e}^{y-\eta }}+1\right)}}={\frac {1}{s+1}}\int _{0}^{\infty }{}dy{\frac {d}{dy}}\left({{y}^{s+1}}\right){\frac {1}{\left({{e}^{y-\eta }}+1\right)}}\\&={\frac {1}{s+1}}\left.\left[\left({{y}^{s+1}}\right){\frac {1}{\left({{e}^{y-\eta }}+1\right)}}\right]\right|_{0}^{\infty }+{\frac {1}{s+1}}\int _{0}^{\infty }{}dy{{y}^{s+1}}{\frac {{e}^{y-\eta }}{{\left({{e}^{y-\eta }}+1\right)}^{2}}}\\&{\frac {1}{s+1}}\left.\left[\left({{y}^{s+1}}\right){\frac {1}{\left({{e}^{y-\eta }}+1\right)}}\right]\right|_{0}^{\infty }=0\\\end{aligned}}

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Γ(s+1)Fs(η):=∫0∞dyys(ey−η+1)=1s+1∫0∞dyddy(ys+1)1(ey−η+1)=1s+1[(ys+1)1(ey−η+1)]|0∞+1s+1∫0∞dyys+1ey−η(ey−η+1)21s+1[(ys+1)1(ey−η+1)]|0∞=0
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