One of the performance metrics calculated in the analysis of telecommunications systems is the ergodic channel capacity, $C\_{\rm erg}$. During one of my studies, I found the expression below for such a metric considering a Rayleigh channel and AWGN.


\begin{align}\label{eq:CAPACIDADEERGODICA}
 C\_{\rm erg} & = 
 \frac{ z^2 }{2 \bar{\gamma} h\_{\rm l}^{2}A\_{0}^{2}\log(2)} {\rm H}\_{3,4}^{4,1}\left[ \frac{1}{\bar{\gamma}h\_{\rm l}^{2}A\_{0}^2} \ \middle\vert \begin{array}{c}
 (-1, 1), (\frac{z^2}{2}, 1), (0, 1) \\
 (0, 1), (\frac{z^2}{2}-1, 1), (-1, 1), (-1, 1)
 \end{array}\right]
\end{align}


To provide more insights into high SNR regimes ($\bar{\gamma} \to \infty$), the asymptotic behavior of this metric is often required.


Therefore, I'd like to know the asymptotic expression of the ergodic capacity, given by the above equation when $\bar{\gamma} \to \infty$.


OBS.: All the other variables can be considered constant.


**UPDATE 10/01/2023**


Additional information on how the previous equation is derived.


The ergodic capacity is found as the expectation of $\log\_2(1+\gamma)$:
\begin{equation}\label{eq:Capacidade}
 C\_{\rm erg} = \int\_{0}^{\infty}\log\_2(1+\gamma)f\_{\Gamma}(\gamma){\rm d}\gamma,
\end{equation}
where
\begin{align}\label{eq:PDFsnr}
 f\_{\Gamma}(\gamma) & = 
 \frac{ z^2}{2 \bar{\gamma} h\_{\rm l}^{2}A\_{0}^{2}} {\rm H}\_{1,2}^{2,0}\left[ \frac{\gamma}{\bar{\gamma} h\_{\rm l}^{2}A\_{0}^2} \ \middle\vert \begin{array}{c}
 (\frac{z^2}{2},1) \\
 (0, 1), (\frac{z^2}{2}-1,1)
 \end{array}\right]
\end{align}
is the PDF of the instantaneous SNR.



Asymptotic analysis of an expression involving a Fox's H function
Your expression can be simplified as follows: As $m=q=4$, the denominator parameter $(0,1)$ cancels with the numerator parameter $(0,1)$. Furthermore, as all linear coefficients are one, the Fox H function reduces to the simpler Meijer G function. I'll denote
$$\tag{1}
\xi = \frac 1 {\bar\gamma h\_1^2 A\_0^2},\quad \zeta = \frac{z^2}{2},
$$
such that
\begin{align}\tag{2a}
 C\_{\rm erg} & = 
 \frac{ \zeta \xi}{\log 2} {\rm G}\_{2,3}^{3,1}\left[ \xi \, \middle\vert \begin{array}{c}
 -1, \zeta \\
 -1, -1, \zeta-1
 \end{array}\right] \\
&= \frac{\zeta}{\log 2} {\rm G}\_{2,3}^{3,1}\left[ \xi \, \middle\vert \begin{array}{c}
 0, \zeta+1 \\
 0, 0, \zeta
 \end{array}\right]\tag{2b}\\
&= \frac{1}{\log 2}\left( 
\frac {\pi \, \Gamma(1-\zeta)} {\sin(\pi\zeta)} \xi^\zeta
-\gamma\_\mathrm E - \zeta^{-1} - \log \xi + \mathcal O(\xi)
\right),\tag{2c}
\end{align}
with [Euler's constant](https://en.wikipedia.org/wiki/Euler%27s_constant) $\gamma\_\mathrm E$, the [Gamma function](https://en.wikipedia.org/wiki/Gamma_function) $\Gamma$, and with the usual big $\mathcal O$ [notation](https://en.wikipedia.org/wiki/Big_O_notation). The calculation was done in *Mathematica*.


