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Display information for equation id:math.2555.52 on revision:2555

* Page found: Das ideale Fermigas (eq math.2555.52)

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Hash: 41926e5c66da2802f86e11e91ba91b28

TeX (original user input): 

\begin{align}

& \Gamma \left( s+1 \right){{F}_{s}}\left( \eta  \right)\approx \frac{{{\left( \eta  \right)}^{s+1}}}{s+1}+\frac{s}{2}{{\left( \eta  \right)}^{s-1}}\frac{{{\pi }^{2}}}{3} \\

& \Gamma \left( s+1 \right){{F}_{s}}\left( \eta  \right)=\frac{{{\left( \eta  \right)}^{s+1}}}{s+1}+\frac{s}{2}{{\left( \eta  \right)}^{s-1}}\frac{{{\pi }^{2}}}{3}+O\left( {{\left( \eta  \right)}^{s-3}} \right) \\

& \Rightarrow {{F}_{s}}\left( \eta  \right)=\frac{1}{\Gamma \left( s+1 \right)}\left[ \frac{{{\left( \eta  \right)}^{s+1}}}{s+1}+\frac{s{{\pi }^{2}}}{6}{{\left( \eta  \right)}^{s-1}}+O\left( {{\left( \eta  \right)}^{s-3}} \right) \right] \\

\end{align}

TeX (checked): 

{\begin{aligned}&\Gamma \left(s+1\right){{F}_{s}}\left(\eta \right)\approx {\frac {{\left(\eta \right)}^{s+1}}{s+1}}+{\frac {s}{2}}{{\left(\eta \right)}^{s-1}}{\frac {{\pi }^{2}}{3}}\\&\Gamma \left(s+1\right){{F}_{s}}\left(\eta \right)={\frac {{\left(\eta \right)}^{s+1}}{s+1}}+{\frac {s}{2}}{{\left(\eta \right)}^{s-1}}{\frac {{\pi }^{2}}{3}}+O\left({{\left(\eta \right)}^{s-3}}\right)\\&\Rightarrow {{F}_{s}}\left(\eta \right)={\frac {1}{\Gamma \left(s+1\right)}}\left[{\frac {{\left(\eta \right)}^{s+1}}{s+1}}+{\frac {s{{\pi }^{2}}}{6}}{{\left(\eta \right)}^{s-1}}+O\left({{\left(\eta \right)}^{s-3}}\right)\right]\\\end{aligned}}

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MathML (44.227 KB / 4.614 KB) :

Γ ⁢ ( s + 1 ) ⁢ F s ⁢ ( η ) ≈ ( η ) s + 1 s + 1 + s 2 ⁢ ( η ) s - 1 ⁢ π 2 3 Γ ⁢ ( s + 1 ) ⁢ F s ⁢ ( η ) = ( η ) s + 1 s + 1 + s 2 ⁢ ( η ) s - 1 ⁢ π 2 3 + O ⁢ ( ( η ) s - 3 ) ⇒ F s ⁢ ( η ) = 1 Γ ⁢ ( s + 1 ) ⁢ [ ( η ) s + 1 s + 1 + s ⁢ π 2 6 ⁢ ( η ) s - 1 + O ⁢ ( ( η ) s - 3 ) ] absent Γ 𝑠 1 subscript 𝐹 𝑠 𝜂 superscript 𝜂 𝑠 1 𝑠 1 𝑠 2 superscript 𝜂 𝑠 1 superscript 𝜋 2 3 absent Γ 𝑠 1 subscript 𝐹 𝑠 𝜂 superscript 𝜂 𝑠 1 𝑠 1 𝑠 2 superscript 𝜂 𝑠 1 superscript 𝜋 2 3 𝑂 superscript 𝜂 𝑠 3 absent ⇒ absent subscript 𝐹 𝑠 𝜂 1 Γ 𝑠 1 delimited-[] superscript 𝜂 𝑠 1 𝑠 1 𝑠 superscript 𝜋 2 6 superscript 𝜂 𝑠 1 𝑂 superscript 𝜂 𝑠 3 {\displaystyle{\displaystyle\begin{aligned} \par&\displaystyle\Gamma\left(s+1% \right){{F}_{s}}\left(\eta\right)\approx\frac{{{\left(\eta\right)}^{s+1}}}{s+1% }+\frac{s}{2}{{\left(\eta\right)}^{s-1}}\frac{{{\pi}^{2}}}{3}\\ \par&\displaystyle\Gamma\left(s+1\right){{F}_{s}}\left(\eta\right)=\frac{{{% \left(\eta\right)}^{s+1}}}{s+1}+\frac{s}{2}{{\left(\eta\right)}^{s-1}}\frac{{{% \pi}^{2}}}{3}+O\left({{\left(\eta\right)}^{s-3}}\right)\\ \par&\displaystyle\Rightarrow{{F}_{s}}\left(\eta\right)=\frac{1}{\Gamma\left(s% +1\right)}\left[\frac{{{\left(\eta\right)}^{s+1}}}{s+1}+\frac{s{{\pi}^{2}}}{6}% {{\left(\eta\right)}^{s-1}}+O\left({{\left(\eta\right)}^{s-3}}\right)\right]\\ \par\end{aligned}}}

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MathML (5.683 KB / 522 B) :

Γ(s+1)Fs(η)≈(η)s+1s+1+s2(η)s−1π23Γ(s+1)Fs(η)=(η)s+1s+1+s2(η)s−1π23+O((η)s−3)⇒Fs(η)=1Γ(s+1)[(η)s+1s+1+sπ26(η)s−1+O((η)s−3)]

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