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Display information for equation id:math.2555.86 on revision:2555
* Page found: Das ideale Fermigas (eq math.2555.86)
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\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}
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<math class="mwe-math-element" xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt"><mtr><mtd></mtd><mtd><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>=</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 mathvariant="normal">Γ</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>s</mi><mo>+</mo><mn>1</mn><mo data-mjx-texclass="CLOSE">)</mo></mrow></mrow></mrow></mfrac></mrow><mstyle displaystyle="true" scriptlevel="0"><munderover><mo texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><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>−</mo><mi>η</mi></mrow></mrow></msup><mo>+</mo><mn>1</mn></mrow></mrow></mfrac></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</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 mathvariant="normal">Γ</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>s</mi><mo>+</mo><mn>1</mn><mo data-mjx-texclass="CLOSE">)</mo></mrow></mrow></mrow></mfrac></mrow><mstyle displaystyle="true" 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data-mjx-texclass="OPEN">(</mo><mi>s</mi><mo>+</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><mstyle displaystyle="true" scriptlevel="0"><munderover><mo texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><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>−</mo><mi>y</mi></mrow></mrow></msup><mo>−</mo><msup><mi>ξ</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mstyle displaystyle="true" scriptlevel="0"><munderover><mo texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><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>−</mo><mn>2</mn><mi>y</mi></mrow></mrow></msup><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mstyle displaystyle="true" scriptlevel="0"><munderover><mo texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><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>−</mo><mi>y</mi></mrow></mrow></msup><mo>=</mo><mi mathvariant="normal">Γ</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>s</mi><mo>+</mo><mn>1</mn><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mstyle displaystyle="true" scriptlevel="0"><munderover><mo texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><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>−</mo><mn>2</mn><mi>y</mi></mrow></mrow></msup><mo>=</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>+</mo><mn>1</mn></mrow></mrow></msup></mrow></mfrac></mrow><mstyle displaystyle="true" scriptlevel="0"><munderover><mo texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><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>−</mo><mi>z</mi></mrow></mrow></msup><mo>=</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>+</mo><mn>1</mn></mrow></mrow></msup></mrow></mfrac></mrow><mi mathvariant="normal">Γ</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>s</mi><mo>+</mo><mn>1</mn><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>⇒</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>≈</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mi>ξ</mi><mo>−</mo><msup><mi>ξ</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mrow data-mjx-texclass="ORD"><mfrac><mrow 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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>−</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>+</mo><mn>1</mn></mrow></mrow></msup></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>
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