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Display information for equation id:math.2550.49 on revision:2550

* Page found: Das ideale Fermigas (eq math.2550.49)

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Das ideale Fermigas Eq: math.2555.49

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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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\begin{aligned} Γ (s + 1) F_{s} (η) = \frac{1}{s + 1} \int_{- η}^{\infty} d x {(x + η)}^{s + 1} \frac{e^{x}}{{(e^{x} + 1)}^{2}} \approx \frac{1}{s + 1} \int_{- \infty}^{\infty} d x {(x + η)}^{s + 1} \frac{e^{x}}{{(e^{x} + 1)}^{2}} + O (e^{- η}) \\ \approx \frac{1}{s + 1} \int_{- \infty}^{\infty} d x {(η)}^{s + 1} \frac{e^{x}}{{(e^{x} + 1)}^{2}} + \int_{- \infty}^{\infty} d x {(η)}^{s} x \frac{e^{x}}{{(e^{x} + 1)}^{2}} + \frac{s}{2} \int_{- \infty}^{\infty} d x {(η)}^{s - 1} x^{2} \frac{e^{x}}{{(e^{x} + 1)}^{2}} \\ = \frac{{(η)}^{s + 1}}{s + 1} \int_{- \infty}^{\infty} d x \frac{e^{x}}{{(e^{x} + 1)}^{2}} + {(η)}^{s} \int_{- \infty}^{\infty} d x \frac{x e^{x}}{{(e^{x} + 1)}^{2}} + \frac{s}{2} {(η)}^{s - 1} \int_{- \infty}^{\infty} d x x^{2} \frac{e^{x}}{{(e^{x} + 1)}^{2}} \end{aligned}

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$Γ$

$s$

$F_{s}$

$η$

$s$

$η$

$x$

$x$

$η$

$s$

$e$

$x$

$e$

$x$

$s$

$x$

$x$

$η$

$s$

$e$

$x$

$e$

$x$

$O$

$e$

$η$

$s$

$x$

$η$

$s$

$e$

$x$

$e$

$x$

$x$

$η$

$s$

$x$

$e$

$x$

$e$

$x$

$s$

$x$

$η$

$s$

$x$

$e$

$x$

$e$

$x$

$η$

$s$

$s$

$x$

$e$

$x$

$e$

$x$

$η$

$s$

$x$

$x$

$e$

$x$

$e$

$x$

$s$

$η$

$s$

$x$

$x$

$e$

$x$

$e$

$x$

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