diff --git a/docs/passive_cable/cable_computation.tex b/docs/passive_cable/cable_computation.tex index 7fa1b18b67e793d25adde9eac17ff7bf0519d92e..39a372196c006c7c67837f039df88c903cb439b4 100644 --- a/docs/passive_cable/cable_computation.tex +++ b/docs/passive_cable/cable_computation.tex @@ -52,7 +52,6 @@ is injected at the left end of the cable from $t=0$, and the initial potential is the reversal potential. \begin{table}[ht] - \label{tbl:rallpack1} \centering \begin{tabular}{lSl} \toprule @@ -67,6 +66,7 @@ is the reversal potential. \bottomrule \end{tabular} \caption{Cable properties for the Rallpack 1 model.} + \label{tbl:rallpack1} \end{table} The potential on a constant-radius passive cable is governed by the PDE @@ -132,7 +132,7 @@ From The boundary conditions \eqref{eq:normcableleft} and \eqref{eq:normcableright} give \begin{align} \label{eq:lapleft} - \frac{\partial G}{\partial x}(0, s) & = \frac{1}{s}, + \frac{\partial G}{\partial x}(0, s) & = \frac{1}{s} \\ \label{eq:lapright} \frac{\partial G}{\partial x}(L, s) & = 0. @@ -144,12 +144,13 @@ Solutions to \eqref{eq:lap} must be of the form \end{equation} where $m=\sqrt{1+s}$. From \eqref{eq:lapleft} and \eqref{eq:lapright}, \begin{align*} - mA(s) - mB(s) & = \frac{1}{s}, + mA(s) - mB(s) & = \frac{1}{s} \\ - mA(s)e^{mL} -mB(s)e^{-mL} & = 0. -\end{align*} and thus + mA(s)e^{mL} -mB(s)e^{-mL} & = 0 +\end{align*} +and thus \begin{align*} - A(s) & = \frac{1}{sm (1-e^{2mL})}, + A(s) & = \frac{1}{sm (1-e^{2mL})} \\ B(s) & = \frac{e^{2mL}}{sm (1-e^{2mL})}. \end{align*} @@ -166,7 +167,7 @@ Consequently, Sufficient conditions for the inverse transform of $G(x,s)$ to exist and be representable in series form are as follows -\parencite[][Theorem 4]{churchill1937}): +\parencite[][Theorem 4]{churchill1937}: \begin{enumerate} \item $G(x, s)$ is analytic in some right half-plane $H$, @@ -213,7 +214,7 @@ $2k\pi\delta/L+\delta^2$ about $s_k$ for $k\geq 1$, which then are separated by circles $|s|=\rho_k$ about the origin for some $\rho_k$ in $(|s_k|, |s_k+1|)$. \textbf{Pole at $s=0$}.\\ -Recalling $m=\sqrt{1+s}$, $m$ and $\sinh mL$ are non-zero in\\ +Recalling $m=\sqrt{1+s}$, $m$ and $\sinh mL$ are non-zero in a neighbourhood of $s=0$, and so the pole is simple and \begin{equation} \begin{aligned} @@ -257,7 +258,7 @@ Consequently the pole is simple and \Res(G; s_k) & = f(x, s_k)/h'(s_k)\\ & = -\frac{2}{s_k L}\frac{\cosh m_k(L-x)}{\cosh m_kL} \\ - & = -\frac{2}{s_k L}\frac{\cosh m_kL\cosh m_kx-\sinh m_kL\sinh m_kL)}{\cosh m_kL} \\ + & = -\frac{2}{s_k L}\frac{\cosh m_kL\cosh m_kx-\sinh m_kL\sinh m_kL}{\cosh m_kL} \\ & = -\frac{2}{s_k L}\cosh m_k x, \end{aligned} \end{equation} @@ -297,10 +298,10 @@ residual. The $a_k$ form an increasing sequence, so \end{aligned} \label{eq:gbar} \end{equation} -Substituting $u=v/\sqrt{t}$ gives the identity +Substituting $u=\sqrt{v/t}$ gives the identity \begin{equation} \int_{a}^\infty \frac{e^{-tu^2}}{u^2}\,du - = \frac{1}{2}\sqrt{t}\int_{a^2t}^\infty e^{-v} v^{\frac{-3}{2}}\,dv + = \frac{1}{2}\sqrt{t}\int_{a^2t}^\infty e^{-tv} v^{\frac{-3}{2}}\,dv = \frac{1}{2}\sqrt{t}\,\Gamma(-\frac{1}{2},a^2 t), \end{equation} where $\Gamma(\alpha, z)$ is the upper incomplete gamma function.