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Display information for equation id:math.2555.93 on revision:2555

* Page found: Das ideale Fermigas (eq math.2555.93)

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TeX (original user input): 

\begin{align}

& U=\frac{\left( 2s+1 \right)}{2}\left( \frac{V}{{{h}^{3}}} \right)4\pi {{\left( 2mkT \right)}^{\frac{3}{2}}}kT\int_{0}^{\infty }{{}}dy\frac{{{y}^{\frac{3}{2}}}}{\left( {{e}^{y-\eta }}+1 \right)}=\frac{\left( 2s+1 \right)}{2}\left( \frac{V}{{{h}^{3}}} \right)4\pi {{\left( 2mkT \right)}^{\frac{3}{2}}}kT\frac{3\sqrt{\pi }}{4}{{F}_{\frac{3}{2}}}\left( \frac{\mu }{kT} \right) \\

& U=V{{N}_{C}}\frac{3}{2}kT{{F}_{\frac{3}{2}}}\left( \frac{\mu }{kT} \right) \\

& U\approx V{{N}_{C}}\frac{3}{2}kT{{e}^{\frac{\mu }{kT}}}\left[ 1-{{e}^{\frac{\mu }{kT}}}\frac{1}{{{2}^{\frac{5}{2}}}} \right] \\

\end{align}

TeX (checked): 

{\begin{aligned}&U={\frac {\left(2s+1\right)}{2}}\left({\frac {V}{{h}^{3}}}\right)4\pi {{\left(2mkT\right)}^{\frac {3}{2}}}kT\int _{0}^{\infty }{}dy{\frac {{y}^{\frac {3}{2}}}{\left({{e}^{y-\eta }}+1\right)}}={\frac {\left(2s+1\right)}{2}}\left({\frac {V}{{h}^{3}}}\right)4\pi {{\left(2mkT\right)}^{\frac {3}{2}}}kT{\frac {3{\sqrt {\pi }}}{4}}{{F}_{\frac {3}{2}}}\left({\frac {\mu }{kT}}\right)\\&U=V{{N}_{C}}{\frac {3}{2}}kT{{F}_{\frac {3}{2}}}\left({\frac {\mu }{kT}}\right)\\&U\approx V{{N}_{C}}{\frac {3}{2}}kT{{e}^{\frac {\mu }{kT}}}\left[1-{{e}^{\frac {\mu }{kT}}}{\frac {1}{{2}^{\frac {5}{2}}}}\right]\\\end{aligned}}

LaTeXML (experimentell; verwendet MathML) rendering

MathML (50.127 KB / 5.454 KB) :

U = ( 2 ⁢ s + 1 ) 2 ⁢ ( V h 3 ) ⁢ 4 ⁢ π ⁢ ( 2 ⁢ m ⁢ k ⁢ T ) 3 2 ⁢ k ⁢ T ⁢ ∫ 0 ∞ 𝑑 y ⁢ y 3 2 ( e y - η + 1 ) = ( 2 ⁢ s + 1 ) 2 ⁢ ( V h 3 ) ⁢ 4 ⁢ π ⁢ ( 2 ⁢ m ⁢ k ⁢ T ) 3 2 ⁢ k ⁢ T ⁢ 3 ⁢ π 4 ⁢ F 3 2 ⁢ ( μ k ⁢ T ) U = V ⁢ N C ⁢ 3 2 ⁢ k ⁢ T ⁢ F 3 2 ⁢ ( μ k ⁢ T ) U ≈ V ⁢ N C ⁢ 3 2 ⁢ k ⁢ T ⁢ e μ k ⁢ T ⁢ [ 1 - e μ k ⁢ T ⁢ 1 2 5 2 ] absent 𝑈 2 𝑠 1 2 𝑉 superscript ℎ 3 4 𝜋 superscript 2 𝑚 𝑘 𝑇 3 2 𝑘 𝑇 superscript subscript 0 differential-d 𝑦 superscript 𝑦 3 2 superscript 𝑒 𝑦 𝜂 1 2 𝑠 1 2 𝑉 superscript ℎ 3 4 𝜋 superscript 2 𝑚 𝑘 𝑇 3 2 𝑘 𝑇 3 𝜋 4 subscript 𝐹 3 2 𝜇 𝑘 𝑇 absent 𝑈 𝑉 subscript 𝑁 𝐶 3 2 𝑘 𝑇 subscript 𝐹 3 2 𝜇 𝑘 𝑇 absent 𝑈 𝑉 subscript 𝑁 𝐶 3 2 𝑘 𝑇 superscript 𝑒 𝜇 𝑘 𝑇 delimited-[] 1 superscript 𝑒 𝜇 𝑘 𝑇 1 superscript 2 5 2 {\displaystyle{\displaystyle\begin{aligned} \par&\displaystyle U=\frac{\left(2% s+1\right)}{2}\left(\frac{V}{{{h}^{3}}}\right)4\pi{{\left(2mkT\right)}^{\frac{% 3}{2}}}kT\int_{0}^{\infty}{{}}dy\frac{{{y}^{\frac{3}{2}}}}{\left({{e}^{y-\eta}% }+1\right)}=\frac{\left(2s+1\right)}{2}\left(\frac{V}{{{h}^{3}}}\right)4\pi{{% \left(2mkT\right)}^{\frac{3}{2}}}kT\frac{3\sqrt{\pi}}{4}{{F}_{\frac{3}{2}}}% \left(\frac{\mu}{kT}\right)\\ \par&\displaystyle U=V{{N}_{C}}\frac{3}{2}kT{{F}_{\frac{3}{2}}}\left(\frac{\mu% }{kT}\right)\\ \par&\displaystyle U\approx V{{N}_{C}}\frac{3}{2}kT{{e}^{\frac{\mu}{kT}}}\left% [1-{{e}^{\frac{\mu}{kT}}}\frac{1}{{{2}^{\frac{5}{2}}}}\right]\\ \par\end{aligned}}}

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MathML (6.073 KB / 654 B) :

U=(2s+1)2(Vh3)4π(2mkT)32kT∫0∞dyy32(ey−η+1)=(2s+1)2(Vh3)4π(2mkT)32kT3π4F32(μkT)U=VNC32kTF32(μkT)U≈VNC32kTeμkT[1−eμkT1252]

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