\right) + z^-_{\bar \Omega} &=&  \frac{gV}{\left( 
    2\pi\right)^3} \int \limits_{\Omega} d^3 p ~ 
e^{-\frac{\varepsilon+\mu}{T}}~ e^{i \phi_{N_{\Omega}}} +
\frac{gV}{\left( 2\pi\right)^3} \int \limits_{\bar \Omega} d^3 p ~ 
e^{-\frac{\varepsilon+\mu}{T}} , \end{aligned}$$ where the single particle energy $\varepsilon = \sqrt{p^2+m^2}$, and $m$, and $g$ are mass and degeneracy factor of $\pi^-$ respectively. Only for momentum states inside the momentum space region $\Omega$ we introduce additionally a Wick-rotated fugacity $\exp \left(i \phi_{N_{\Omega}} \right)$. For the positive and neutral pion (which we do not want to detect in our example) we write: $$\begin{aligned}
z^+ ~=~  \frac{gV}{\left( 2\pi\right)^3} \int  d^3 p ~
e^{-\frac{\varepsilon-\mu}{T}}~, 
\qquad \qquad \textrm{and} \qquad \qquad 
z^0 ~=~  \frac{gV}{\left( 2\pi\right)^3} \int  d^3 p ~
e^{-\frac{\varepsilon}{T}}~.\end{aligned}$$ The value of the single particle partition function, for instance of the neutral pion, is given by: $$\label{aveN}
z^0=\langle N^0 \rangle = \frac{gV}{2\pi} m^2 T K_2 \left( \frac{m}{T}\right).$$ For the sake of simplicity we assume equal masses for all pions. To obtain the GCE multiplicity distribution for $N_{\Omega}$ in a momentum space segment $\Omega$ we use the Fourier integral over the generalized GCE partition function $ \mathcal{Z} \left( \phi_{N_{\Omega}}\right)=\exp \left[ z^-_{\Omega}
  \left( \phi_{N_{\Omega}} \right) + z^-_{\bar{ \Omega}}  + z^+ + z^0 \right] $, normalized by the GCE partition function: $$\begin{aligned}
\label{GCEPDF}
