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Display information for equation id:math.2383.57 on revision:2383

* Page found: Wahrscheinlichkeitsbegriff (eq math.2383.57)

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Wahrscheinlichkeitsbegriff Eq: math.2386.57

Hash: d9a615da31946cf9c38bfdd7eef192c2

TeX (original user input):

\begin{align}
& Z(a)=\left\langle {{e}^{ax}} \right\rangle =\int_{{}}^{{}}{d{{x}_{1}}d{{x}_{2}}\rho \left( {{x}_{1}} \right)}\rho \left( {{x}_{2}} \right){{e}^{{{a}_{1}}{{x}_{1}}}}{{e}^{{{a}_{2}}{{x}_{2}}}}=\left\langle {{e}^{{{a}_{1}}{{x}_{1}}}} \right\rangle \left\langle {{e}^{{{a}_{2}}{{x}_{2}}}} \right\rangle  \\
& \Rightarrow \Gamma \left( a \right)=\ln Z(a)=\ln \left\langle {{e}^{{{a}_{1}}{{x}_{1}}}} \right\rangle +\ln \left\langle {{e}^{{{a}_{2}}{{x}_{2}}}} \right\rangle =\Gamma \left( {{a}_{1}} \right)+\Gamma \left( {{a}_{2}} \right) \\
& {{\left. \frac{{{\partial }^{n}}}{\partial {{a}^{n}}}\Gamma \left( a \right) \right|}_{a=0}}\Rightarrow {{\left\langle {{\left( {{x}_{1}}+{{x}_{2}} \right)}^{n}} \right\rangle }_{C}}={{\left\langle {{x}^{n}} \right\rangle }_{C}}={{\left\langle {{x}_{1}}^{n} \right\rangle }_{C}}+{{\left\langle {{x}_{2}}^{n} \right\rangle }_{C}} \\
\end{align}

TeX (checked):

{\begin{aligned}&Z(a)=\left\langle {{e}^{ax}}\right\rangle =\int _{}^{}{d{{x}_{1}}d{{x}_{2}}\rho \left({{x}_{1}}\right)}\rho \left({{x}_{2}}\right){{e}^{{{a}_{1}}{{x}_{1}}}}{{e}^{{{a}_{2}}{{x}_{2}}}}=\left\langle {{e}^{{{a}_{1}}{{x}_{1}}}}\right\rangle \left\langle {{e}^{{{a}_{2}}{{x}_{2}}}}\right\rangle \\&\Rightarrow \Gamma \left(a\right)=\ln Z(a)=\ln \left\langle {{e}^{{{a}_{1}}{{x}_{1}}}}\right\rangle +\ln \left\langle {{e}^{{{a}_{2}}{{x}_{2}}}}\right\rangle =\Gamma \left({{a}_{1}}\right)+\Gamma \left({{a}_{2}}\right)\\&{{\left.{\frac {{\partial }^{n}}{\partial {{a}^{n}}}}\Gamma \left(a\right)\right|}_{a=0}}\Rightarrow {{\left\langle {{\left({{x}_{1}}+{{x}_{2}}\right)}^{n}}\right\rangle }_{C}}={{\left\langle {{x}^{n}}\right\rangle }_{C}}={{\left\langle {{x}_{1}}^{n}\right\rangle }_{C}}+{{\left\langle {{x}_{2}}^{n}\right\rangle }_{C}}\\\end{aligned}}

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\begin{aligned} Z (a) = ⟨ e^{a x} ⟩ = \int_{}^{} d x_{1} d x_{2} ρ (x_{1}) ρ (x_{2}) e^{a_{1} x_{1}} e^{a_{2} x_{2}} = ⟨ e^{a_{1} x_{1}} ⟩ ⟨ e^{a_{2} x_{2}} ⟩ \\ \Rightarrow Γ (a) = \ln Z (a) = \ln ⟨ e^{a_{1} x_{1}} ⟩ + \ln ⟨ e^{a_{2} x_{2}} ⟩ = Γ (a_{1}) + Γ (a_{2}) \\ {\frac{\partial^{n}}{\partial a^{n}} Γ (a) |}_{a = 0} \Rightarrow {⟨ {(x_{1} + x_{2})}^{n} ⟩}_{C} = {⟨ x^{n} ⟩}_{C} = {⟨ {x_{1}}^{n} ⟩}_{C} + {⟨ {x_{2}}^{n} ⟩}_{C} \end{aligned}

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Identifiers

$Z$

$a$

$e$

$a$

$x$

$x_{1}$

$d$

$x_{2}$

$ρ$

$x_{1}$

$ρ$

$x_{2}$

$e$

$a_{1}$

$x_{1}$

$e$

$a_{2}$

$x_{2}$

$e$

$a_{1}$

$x_{1}$

$e$

$a_{2}$

$x_{2}$

$Γ$

$a$

$Z$

$a$

$e$

$a_{1}$

$x_{1}$

$e$

$a_{2}$

$x_{2}$

$Γ$

$a_{1}$

$Γ$

$a_{2}$

$n$

$a$

$n$

$Γ$

$a$

$a$

$x_{1}$

$x_{2}$

$n$

$C$

$x$

$n$

$C$

$x_{1}$

$n$

$C$

$x_{2}$

$n$

$C$

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