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Display information for equation id:math.2552.86 on revision:2552

* Page found: Das ideale Fermigas (eq math.2552.86)

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Das ideale Fermigas Eq: math.2555.86

Hash: 4f1a407ae9808503b3a81da5e42d56f4

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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\begin{matrix} F_{s} (η) = \frac{1}{Γ (s + 1)} \int_{0}^{\infty} d y \frac{y^{s}}{e^{y - η} + 1} \\ = \frac{1}{Γ (s + 1)} \int_{0}^{\infty} d y y^{s} \frac{ξ e^{- y}}{1 + ξ e^{- y}} \approx \frac{1}{Γ (s + 1)} [ξ \int_{0}^{\infty} d y y^{s} e^{- y} - ξ^{2} \int_{0}^{\infty} d y y^{s} e^{- 2 y} + ....] \\ \int_{0}^{\infty} d y y^{s} e^{- y} = Γ (s + 1) \\ \int_{0}^{\infty} d y y^{s} e^{- 2 y} = \frac{1}{2^{s + 1}} \int_{0}^{\infty} d z z^{s} e^{- z} = \frac{1}{2^{s + 1}} Γ (s + 1) \\ \Rightarrow F_{s} (η) \approx [ξ - ξ^{2} \frac{1}{2^{s + 1}} + ....] \approx [ξ - ξ^{2} \frac{1}{2^{s + 1}}] = e^{\frac{μ}{k T}} [1 - e^{\frac{μ}{k T}} \frac{1}{2^{s + 1}}] \end{matrix}

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mathvariant="normal">∞</mi></mrow></msubsup><mi>d</mi><mi>y</mi><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msup><mi>y</mi><mrow data-mjx-texclass="ORD"><mi>s</mi></mrow></msup></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>y</mi><mo stretchy="false">−</mo><mi>η</mi></mrow></mrow></msup><mo stretchy="false">+</mo><mn>1</mn></mrow></mrow></mfrac></mrow></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mo stretchy="false">=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>Γ</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>s</mi><mo stretchy="false">+</mo><mn>1</mn><mo data-mjx-texclass="CLOSE">)</mo></mrow></mrow></mrow></mfrac></mrow><msubsup><mo stretchy="false">∫</mo><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>d</mi><mi>y</mi><msup><mi>y</mi><mrow data-mjx-texclass="ORD"><mi>s</mi></mrow></msup><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>ξ</mi><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo stretchy="false">−</mo><mi>y</mi></mrow></mrow></msup></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>1</mn><mo stretchy="false">+</mo><mi>ξ</mi><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo stretchy="false">−</mo><mi>y</mi></mrow></mrow></msup></mrow></mrow></mfrac></mrow><mo stretchy="false">≈</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>Γ</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>s</mi><mo stretchy="false">+</mo><mn>1</mn><mo data-mjx-texclass="CLOSE">)</mo></mrow></mrow></mrow></mfrac></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mi>ξ</mi><msubsup><mo stretchy="false">∫</mo><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>d</mi><mi>y</mi><msup><mi>y</mi><mrow data-mjx-texclass="ORD"><mi>s</mi></mrow></msup><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo stretchy="false">−</mo><mi>y</mi></mrow></mrow></msup><mo stretchy="false">−</mo><msup><mi>ξ</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><msubsup><mo stretchy="false">∫</mo><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>d</mi><mi>y</mi><msup><mi>y</mi><mrow data-mjx-texclass="ORD"><mi>s</mi></mrow></msup><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo stretchy="false">−</mo><mn>2</mn><mi>y</mi></mrow></mrow></msup><mo stretchy="false">+</mo><mi>....</mi><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><msubsup><mo stretchy="false">∫</mo><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>d</mi><mi>y</mi><msup><mi>y</mi><mrow data-mjx-texclass="ORD"><mi>s</mi></mrow></msup><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo stretchy="false">−</mo><mi>y</mi></mrow></mrow></msup><mo stretchy="false">=</mo><mi>Γ</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>s</mi><mo stretchy="false">+</mo><mn>1</mn><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><msubsup><mo stretchy="false">∫</mo><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>d</mi><mi>y</mi><msup><mi>y</mi><mrow data-mjx-texclass="ORD"><mi>s</mi></mrow></msup><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo stretchy="false">−</mo><mn>2</mn><mi>y</mi></mrow></mrow></msup><mo stretchy="false">=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><msup><mn>2</mn><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>s</mi><mo stretchy="false">+</mo><mn>1</mn></mrow></mrow></msup></mrow></mfrac></mrow><msubsup><mo stretchy="false">∫</mo><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>d</mi><mi>z</mi><msup><mi>z</mi><mrow data-mjx-texclass="ORD"><mi>s</mi></mrow></msup><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo stretchy="false">−</mo><mi>z</mi></mrow></mrow></msup><mo stretchy="false">=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><msup><mn>2</mn><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>s</mi><mo stretchy="false">+</mo><mn>1</mn></mrow></mrow></msup></mrow></mfrac></mrow><mi>Γ</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>s</mi><mo stretchy="false">+</mo><mn>1</mn><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mo stretchy="false">⇒</mo><msub><mi>F</mi><mrow data-mjx-texclass="ORD"><mi>s</mi></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>η</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">≈</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mi>ξ</mi><mo stretchy="false">−</mo><msup><mi>ξ</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><msup><mn>2</mn><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>s</mi><mo stretchy="false">+</mo><mn>1</mn></mrow></mrow></msup></mrow></mfrac></mrow><mo stretchy="false">+</mo><mi>....</mi><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo stretchy="false">≈</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mi>ξ</mi><mo stretchy="false">−</mo><msup><mi>ξ</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><msup><mn>2</mn><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>s</mi><mo stretchy="false">+</mo><mn>1</mn></mrow></mrow></msup></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo stretchy="false">=</mo><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>μ</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>k</mi><mi>T</mi></mrow></mrow></mfrac></mrow></mrow></msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mn>1</mn><mo stretchy="false">−</mo><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>μ</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>k</mi><mi>T</mi></mrow></mrow></mfrac></mrow></mrow></msup><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><msup><mn>2</mn><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>s</mi><mo stretchy="false">+</mo><mn>1</mn></mrow></mrow></msup></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd></mtr></mtable></mstyle></mrow></math>

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Identifiers

$F_{s}$

$η$

$Γ$

$s$

$y$

$y$

$s$

$e$

$y$

$η$

$Γ$

$s$

$y$

$y$

$s$

$ξ$

$e$

$y$

$ξ$

$e$

$y$

$Γ$

$s$

$ξ$

$y$

$y$

$s$

$e$

$y$

$ξ$

$y$

$y$

$s$

$e$

$y$

$y$

$y$

$s$

$e$

$y$

$Γ$

$s$

$y$

$y$

$s$

$e$

$y$

$s$

$z$

$z$

$s$

$e$

$z$

$s$

$Γ$

$s$

$F_{s}$

$η$

$ξ$

$ξ$

$s$

$ξ$

$ξ$

$s$

$e$

$μ$

$k$

$T$

$e$

$μ$

$k$

$T$

$s$

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