We now have $$\begin{aligned}
\frac{E_b}{N_0}\bigg|_{{{\sf{R}}}=
0}&=\lim_{\zeta\rightarrow0}\frac{\frac{{\bar{P}}}{N_0} \zeta}{{{\sf{R}}}_E(\zeta)} = \frac{\frac{{\bar{P}}}{N_0}}{\dot{{{\sf{R}}}}_E(0)}=\frac{-\frac{\theta
T{\bar{P}}}{N_0}}{\log_e\big(1-P\{z>{{\alpha^{\ast}_{\text{opt}}}}\}(1-e^{-\theta
T{{{r^{\ast}_{\text{opt}}}}}})\big)}=\frac{-\delta\log_e2}{\log_e\xi}\label{eq:ebminb}\end{aligned}$$ where $\dot{{{\sf{R}}}}_E(0)$ is the derivative of ${{\sf{R}}}_E$ with respect to $\zeta$ at $\zeta = 0$, $\delta=\frac{\theta T{\bar{P}}}{N_0\log_e2}$, and $\xi=1-P\{z>{{\alpha^{\ast}_{\text{opt}}}}\}(1-e^{-\delta{{\alpha^{\ast}_{\text{opt}}}}})$. Therefore, we prove (\[eq:ebminwb\]). Note that the second derivative $\ddot{{{\sf{R}}}}_E(0)$, required in the computation of the wideband slope $\mathcal{S}_0$, can be obtained from $$\begin{aligned}
\ddot{{{\sf{R}}}}_E(0)&=\lim_{\zeta\rightarrow0}2\frac{{{\sf{R}}}_E(\zeta)-\dot{{{\sf{R}}}}_E(0)\zeta}{\zeta^2}\nonumber\\
&=\lim_{\zeta\rightarrow0}2\frac{1}{\zeta}\Big(-\frac{1}{\theta
T}\log_e\big(1-P\{z>{\alpha_{\text{opt}}}\}\big(1-e^{-\theta T{r_{\text{opt}}}}\big)\big)
+\frac{1}{\theta T}\log_e\big(1-P\{z>{{\alpha^{\ast}_{\text{opt}}}}\}(1-e^{-\theta
T{{{r^{\ast}_{\text{opt}}}}}})\big)\Big)\nonumber\\
&=\lim_{\zeta\rightarrow0}-\frac{2}{\theta
T}\frac{\big(p_z({\alpha_{\text{opt}}}){\dot{\alpha}_{\text{opt}}}(\zeta)(1-e^{-\theta
T{r_{\text{opt}}}})-P\{z>{\alpha_{\text{opt}}}\}\theta Te^{-\theta
T{r_{\text{opt}}}}{\dot{r}_{\text{opt}}}(\zeta)\big)}{1-P\{z>{\alpha_{\text{opt}}}\}\big(1-e^{-\theta
T{r_{\text{opt}}}}\big)}\label{eq:ddotproof1}\\
&=-\frac{2}{\theta T}\frac{\big(p_z({{\alpha^{\ast}_{\text{opt}}}}){\dot{\alpha}_{\text{opt}}}(0)(1-e^{-\theta
T{{{r^{\ast}_{\text{opt}}}}}})-P\{z>{{\alpha^{\ast}_{\text{opt}}}}\}\theta Te^{-\theta
T{{{r^{\ast}_{\text{opt}}}}}}{\dot{r}_{\text{opt}}}(0)\big)}{1-P\{z>{{\alpha^{\ast}_{\text{opt}}}}\}\big(1-e^{-\theta T{{{r^{\ast}_{\text{opt}}}}}}\big)}\label{eq:ddotproof2}\end{aligned}$$ where ${{{r^{\ast}_{\text{opt}}}}}=\frac{{\bar{P}}{{\alpha^{\ast}_{\text{opt}}}}}{N_0\log_e2}$. Above, (\[eq:ddotproof1\]) and (\[eq:ddotproof2\]) follow by using L’Hospital’s Rule and applying Leibniz Integral Rule [@Protter]. Next, we derive an equality satisfied by ${{\alpha^{\ast}_{\text{opt}}}}$. Consider the objective function in (\[eq:specec\]) $$\begin{gathered}
\label{eq:objfunc}
-\frac{1}{\theta TB}\log_e\big(1-P\{z>\alpha\}(1-e^{-\theta
Tr})\big).\end{gathered}$$ It can easily be seen that both as $r \to 0$ and $r \to \infty$, this objective function approaches zero[^7]. Hence, (\[eq:objfunc\]) is maximized at a finite and nonzero value of $r$ at which the derivative of (\[eq:objfunc\]) with respect to $r$ is zero. Differentiating (\[eq:objfunc\]) with respect to $r$ and making it equal to zero leads to the equality that needs to be satisfied at the optimal value ${r_{\text{opt}}}$: $$\label{eq:conditionb}
\frac{2^{{r_{\text{opt}}}\zeta}p_z({\alpha_{\text{opt}}})N_0\log_e2}{{\bar{P}}}(1-e^{-\theta
T{r_{\text{opt}}}})=\theta Te^{-\theta T{r_{\text{opt}}}}P\{z>{\alpha_{\text{opt}}}\}$$ where $\zeta = 1/B$. For given $\theta$, as the bandwidth increases (i.e., $\zeta\rightarrow0$), ${r_{\text{opt}}}\to {{{r^{\ast}_{\text{opt}}}}}$. Clearly, 