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* Page found: Das ideale Bosegas (eq math.2557.0)

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

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Y = ∑ N 1 ⁢ … ⁢ N l = 0 ∞ exp ⁡ ( - β ⁢ ∑ j = 1 l ( N j ⁢ E j - μ ⁢ N j ) ) = ∏ j = 1 l ( ∑ N j = 0 ∞ exp ⁡ ( - β ⁢ ( N j ⁢ E j - μ ⁢ N j ) ) ) = ∏ j = 1 l ( ∑ N j = 0 ∞ t j N j ) t j := exp ⁡ ( - β ⁢ ( E j - μ ) ) Y = ∏ j = 1 l 1 1 - t j = ∏ j = 1 l Y j missing-subexpression 𝑌 superscript subscript subscript 𝑁 1 … subscript 𝑁 𝑙 0 𝛽 superscript subscript 𝑗 1 𝑙 subscript 𝑁 𝑗 subscript 𝐸 𝑗 𝜇 subscript 𝑁 𝑗 superscript subscript product 𝑗 1 𝑙 superscript subscript subscript 𝑁 𝑗 0 𝛽 subscript 𝑁 𝑗 subscript 𝐸 𝑗 𝜇 subscript 𝑁 𝑗 missing-subexpression absent superscript subscript product 𝑗 1 𝑙 superscript subscript subscript 𝑁 𝑗 0 superscript subscript 𝑡 𝑗 subscript 𝑁 𝑗 missing-subexpression assign subscript 𝑡 𝑗 𝛽 subscript 𝐸 𝑗 𝜇 missing-subexpression 𝑌 superscript subscript product 𝑗 1 𝑙 1 1 subscript 𝑡 𝑗 superscript subscript product 𝑗 1 𝑙 subscript 𝑌 𝑗 {\displaystyle{\displaystyle\begin{aligned} &\displaystyle Y=\sum\limits_{{{N}% _{1}}...{{N}_{l}}=0}^{\infty}{{}}\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}^{\infty}{{}}\exp\left(-\beta\left({{N}_{j}}{{E% }_{j}}-\mu{{N}_{j}}\right)\right)\right)\\ &\displaystyle=\prod\limits_{j=1}^{l}{{}}\left(\sum\limits_{{{N}_{j}}=0}^{% \infty}{{}}{{t}_{j}}^{{{N}_{j}}}\right)\\ &\displaystyle{{t}_{j}}:=\exp\left(-\beta\left({{E}_{j}}-\mu\right)\right)\\ &\displaystyle Y=\prod\limits_{j=1}^{l}{{}}\frac{1}{1-{{t}_{j}}}=\prod\limits_% {j=1}^{l}{{}}{{Y}_{j}}\\ \end{aligned}}}

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Y=∑N1...Nl=0∞exp⁡(−β∑j=1l(NjEj−μNj))=∏j=1l(∑Nj=0∞exp⁡(−β(NjEj−μNj)))=∏j=1l(∑Nj=0∞tjNj)tj:=exp⁡(−β(Ej−μ))Y=∏j=1l11−tj=∏j=1lYj

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