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Display information for equation id:math.2578.6 on revision:2578
* Page found: Ferromagnetismus (eq math.2578.6)
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Hash: 40dfbd1977940ca43dc0b44bffd7836e
TeX (original user input):
\begin{align}
& Z=\sum\limits_{{{m}_{l}}=-l}^{l}{{}}\exp \left( \beta \mu B{{m}_{l}} \right) \\
& \nu :={{m}_{l}}+l \\
& \Rightarrow Z=\exp \left( -\beta \mu Bl \right)\sum\limits_{\nu =0}^{2l}{{}}{{\left( \exp \left( \beta \mu B \right) \right)}^{\nu }}=\exp \left( -\beta \mu Bl \right)\frac{\exp \left( \beta \mu B\left( 2l+1 \right) \right)-1}{\exp \left( \beta \mu B \right)-1}=\frac{\sinh \left( \beta \mu B\left( l+\frac{1}{2} \right) \right)}{\sinh \left( \frac{1}{2}\beta \mu B \right)} \\
\end{align}
TeX (checked):
{\begin{aligned}&Z=\sum \limits _{{{m}_{l}}=-l}^{l}{}\exp \left(\beta \mu B{{m}_{l}}\right)\\&\nu :={{m}_{l}}+l\\&\Rightarrow Z=\exp \left(-\beta \mu Bl\right)\sum \limits _{\nu =0}^{2l}{}{{\left(\exp \left(\beta \mu B\right)\right)}^{\nu }}=\exp \left(-\beta \mu Bl\right){\frac {\exp \left(\beta \mu B\left(2l+1\right)\right)-1}{\exp \left(\beta \mu B\right)-1}}={\frac {\sinh \left(\beta \mu B\left(l+{\frac {1}{2}}\right)\right)}{\sinh \left({\frac {1}{2}}\beta \mu B\right)}}\\\end{aligned}}
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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><mi>Z</mi><mo>=</mo><munderover><mo form="prefix" texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><msub><mi>m</mi><mrow data-mjx-texclass="ORD"><mi>l</mi></mrow></msub><mo>=</mo><mo>−</mo><mi>l</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>l</mi></mrow></munderover><mi>exp</mi><mo>⁡</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>β</mi><mi>μ</mi><mi>B</mi><msub><mi>m</mi><mrow data-mjx-texclass="ORD"><mi>l</mi></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mi>ν</mi><mi>:</mi><mo>=</mo><msub><mi>m</mi><mrow data-mjx-texclass="ORD"><mi>l</mi></mrow></msub><mo>+</mo><mi>l</mi></mtd></mtr><mtr><mtd></mtd><mtd><mo>⇒</mo><mi>Z</mi><mo>=</mo><mi>exp</mi><mo>⁡</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mo>−</mo><mi>β</mi><mi>μ</mi><mi>B</mi><mi>l</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><munderover><mo form="prefix" texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>ν</mi><mo>=</mo><mn>0</mn></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>2</mn><mi>l</mi></mrow></mrow></munderover><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>exp</mi><mo>⁡</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>β</mi><mi>μ</mi><mi>B</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mi>ν</mi></mrow></msup><mo>=</mo><mi>exp</mi><mo>⁡</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mo>−</mo><mi>β</mi><mi>μ</mi><mi>B</mi><mi>l</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>exp</mi><mo>⁡</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>β</mi><mi>μ</mi><mi>B</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mn>2</mn><mi>l</mi><mo>+</mo><mn>1</mn><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>−</mo><mn>1</mn></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>exp</mi><mo>⁡</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>β</mi><mi>μ</mi><mi>B</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>−</mo><mn>1</mn></mrow></mrow></mfrac></mrow><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>sinh</mi><mo>⁡</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>β</mi><mi>μ</mi><mi>B</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>l</mi><mo>+</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>sinh</mi><mo>⁡</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></mfrac></mrow><mi>β</mi><mi>μ</mi><mi>B</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow></mrow></mrow></mfrac></mrow></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>
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