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Display information for equation id:math.2555.5 on revision:2555

* Page found: Das ideale Fermigas (eq math.2555.5)

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Das ideale Fermigas Eq: math.2555.5

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TeX (original user input):

\begin{align}

& Y=\sum\limits_{{{N}_{1}}...{{N}_{l}}=0}^{1}{{}}\exp \left( -\beta \sum\limits_{j=1}^{l}{{}}\left( {{N}_{j}}{{E}_{j}}-\mu {{N}_{j}} \right) \right)=\prod\limits_{j=1}^{l}{{}}\left( \sum\limits_{{{N}_{j}}=0}^{1}{{}}\exp \left( -\beta \left( {{N}_{j}}{{E}_{j}}-\mu {{N}_{j}} \right) \right) \right) \\

& =\prod\limits_{j=1}^{l}{{}}\left( \sum\limits_{{{N}_{j}}=0}^{1}{{}}{{t}_{j}}^{{{N}_{j}}} \right) \\

& {{t}_{j}}:=\exp \left( -\beta \left( {{E}_{j}}-\mu  \right) \right) \\

& Y=\prod\limits_{j=1}^{l}{{}}\left( 1+{{t}_{j}} \right)=\prod\limits_{j=1}^{l}{{}}{{Y}_{j}} \\

\end{align}

TeX (checked):

{\begin{aligned}&Y=\sum \limits _{{{N}_{1}}...{{N}_{l}}=0}^{1}{}\exp \left(-\beta \sum \limits _{j=1}^{l}{}\left({{N}_{j}}{{E}_{j}}-\mu {{N}_{j}}\right)\right)=\prod \limits _{j=1}^{l}{}\left(\sum \limits _{{{N}_{j}}=0}^{1}{}\exp \left(-\beta \left({{N}_{j}}{{E}_{j}}-\mu {{N}_{j}}\right)\right)\right)\\&=\prod \limits _{j=1}^{l}{}\left(\sum \limits _{{{N}_{j}}=0}^{1}{}{{t}_{j}}^{{N}_{j}}\right)\\&{{t}_{j}}:=\exp \left(-\beta \left({{E}_{j}}-\mu \right)\right)\\&Y=\prod \limits _{j=1}^{l}{}\left(1+{{t}_{j}}\right)=\prod \limits _{j=1}^{l}{}{{Y}_{j}}\\\end{aligned}}

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MathML (4.771 KB / 529 B) :

\begin{aligned} Y = \sum_{N_{1} . . . N_{l} = 0}^{1} \exp (- β \sum_{j = 1}^{l} (N_{j} E_{j} - μ N_{j})) = \prod_{j = 1}^{l} (\sum_{N_{j} = 0}^{1} \exp (- β (N_{j} E_{j} - μ N_{j}))) \\ = \prod_{j = 1}^{l} (\sum_{N_{j} = 0}^{1} {t_{j}}^{N_{j}}) \\ t_{j} : = \exp (- β (E_{j} - μ)) \\ Y = \prod_{j = 1}^{l} (1 + t_{j}) = \prod_{j = 1}^{l} Y_{j} \end{aligned}

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Translations to Computer Algebra Systems

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Identifiers

$Y$

$N_{1}$

$N_{l}$

$β$

$j$

$l$

$N_{j}$

$E_{j}$

$μ$

$N_{j}$

$j$

$l$

$N_{j}$

$β$

$N_{j}$

$E_{j}$

$μ$

$N_{j}$

$j$

$l$

$N_{j}$

$t_{j}$

$N_{j}$

$t_{j}$

$β$

$E_{j}$

$μ$

$Y$

$j$

$l$

$t_{j}$

$j$

$l$

$Y_{j}$

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