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Display information for equation id:math.2563.20 on revision:2563
* Page found: Das Photonengas im Strahlungshohlraum (eq math.2563.20)
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Hash: 75d31515ac7f2b894fcacc28d9157052
TeX (original user input):
\begin{align}
& U\left( T \right):=V\int_{{}}^{{}}{{}}d\nu \frac{1}{V}D\left( \nu \right)h\nu \left\langle {{N}_{\nu }} \right\rangle =\frac{8\pi hV}{{{c}^{3}}}\int_{0}^{\infty }{{}}d\nu \frac{{{\nu }^{3}}}{{{e}^{\frac{h\nu }{kT}}}-1} \\
& =\frac{8\pi V}{{{\left( ch \right)}^{3}}}{{\left( kT \right)}^{4}}\int_{0}^{\infty }{{}}dx\frac{{{x}^{3}}}{{{e}^{x}}-1} \\
& \int_{0}^{\infty }{{}}dx\frac{{{x}^{3}}}{{{e}^{x}}-1}=\frac{{{\pi }^{4}}}{15} \\
& U\left( T \right)=\frac{8{{\pi }^{5}}V}{15{{\left( ch \right)}^{3}}}{{\left( kT \right)}^{4}} \\
\end{align}
TeX (checked):
{\begin{aligned}&U\left(T\right):=V\int _{}^{}{}d\nu {\frac {1}{V}}D\left(\nu \right)h\nu \left\langle {{N}_{\nu }}\right\rangle ={\frac {8\pi hV}{{c}^{3}}}\int _{0}^{\infty }{}d\nu {\frac {{\nu }^{3}}{{{e}^{\frac {h\nu }{kT}}}-1}}\\&={\frac {8\pi V}{{\left(ch\right)}^{3}}}{{\left(kT\right)}^{4}}\int _{0}^{\infty }{}dx{\frac {{x}^{3}}{{{e}^{x}}-1}}\\&\int _{0}^{\infty }{}dx{\frac {{x}^{3}}{{{e}^{x}}-1}}={\frac {{\pi }^{4}}{15}}\\&U\left(T\right)={\frac {8{{\pi }^{5}}V}{15{{\left(ch\right)}^{3}}}}{{\left(kT\right)}^{4}}\\\end{aligned}}
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<math xmlns="http://www.w3.org/1998/Math/MathML" class="mwe-math-element mwe-math-element-inline"><mrow data-mjx-texclass="ORD"><mstyle displaystyle="true" scriptlevel="0"><mtable displaystyle="true"><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mi>U</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>T</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">:=</mo><mi>V</mi><msubsup><mo stretchy="false">∫</mo><mrow data-mjx-texclass="ORD"></mrow><mrow data-mjx-texclass="ORD"></mrow></msubsup><mi>d</mi><mi>ν</mi><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>V</mi></mrow></mfrac></mrow><mi>D</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>ν</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mi>h</mi><mi>ν</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msub><mi>N</mi><mrow data-mjx-texclass="ORD"><mi>ν</mi></mrow></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mo stretchy="false">=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>8</mn><mi>π</mi><mi>h</mi><mi>V</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><msup><mi>c</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></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>ν</mi><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msup><mi>ν</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></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"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>h</mi><mi>ν</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>k</mi><mi>T</mi></mrow></mrow></mfrac></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"><mrow data-mjx-texclass="ORD"><mn>8</mn><mi>π</mi><mi>V</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>c</mi><mi>h</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup></mrow></mfrac></mrow><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>k</mi><mi>T</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mn>4</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>x</mi><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msup><mi>x</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mi>x</mi></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"><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>x</mi><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msup><mi>x</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mi>x</mi></mrow></msup><mo stretchy="false">−</mo><mn>1</mn></mrow></mrow></mfrac></mrow><mo stretchy="false">=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msup><mi>π</mi><mrow data-mjx-texclass="ORD"><mn>4</mn></mrow></msup></mrow><mrow data-mjx-texclass="ORD"><mn>15</mn></mrow></mfrac></mrow></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mi>U</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>T</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>8</mn><msup><mi>π</mi><mrow data-mjx-texclass="ORD"><mn>5</mn></mrow></msup><mi>V</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>15</mn><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>c</mi><mi>h</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup></mrow></mrow></mfrac></mrow><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>k</mi><mi>T</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mn>4</mn></mrow></msup></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd></mtr></mtable></mstyle></mrow></math>
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