Finalize cable computation document.

docs/passive_cable/cable.bib

 @article{bhalla1992,
     title = {Rallpacks: a set of benchmarks for neuronal simulators},
+    author = {Upinder S. Bhalla and David H. Bilitch and James M. Bower},
     journal = {Trends in Neurosciences},
     volume = {15},
     number = {11},
     pages = {453--458},
     year = {1992},
     month = {11},
-    author = {Upinder S. Bhalla and David H. Bilitch and James M. Bower},
     doi = {10.1016/0166-2236(92)90009-W}
 }
+
+@article{churchill1937,
+    title = {The inversion of the Laplace transformation by a direct expansion in series and its application to boundary-value problems},
+    author = {R. V. Churchill},
+    journal = {Mathematische Zeitschrift},
+    volume = {42},
+    number = {1},
+    pages = {567--579},
+    year = {1937},
+    doi = {10.1007/BF01160095}
+}
+
+@article{borwein2009,
+    title = {Uniform bounds for the complementary incomplete gamma function},
+    author = {Jonathan M Borwein and O-Yeat Chan},
+    journal = {Mathematical Inequalities and Applications},
+    volume = {12},
+    number = {1},
+    pages = {115--121},
+    year = {2009},
+    doi = {10.7153/mia-12-10}
+}

docs/passive_cable/cable_computation.tex

 \documentclass[parskip=half]{scrartcl}
 
 \usepackage{amsmath}
+\usepackage{amssymb}
 \usepackage[citestyle=authoryear-icomp,bibstyle=authoryear]{biblatex}
 \usepackage{booktabs}
 \usepackage[style=british]{csquotes}
@@ -17,6 +18,8 @@
 \addbibresource{cable.bib}
 
 \title{Analytic solution to passive cable model}
+\author{Sam Yates}
+\date{October 20, 2016}
 
 \begin{document}
 
@@ -24,7 +27,7 @@
 
 \section{Background}
 
-The Rallpack suite \cite{bhalla1992} is a set of three tests for cable-based neuronal
+The Rallpack suite \parencite{bhalla1992} is a set of three tests for cable-based neuronal
 simulators, for validation and benchmarking. Validation data is provided with the
 distribution of the suite, which at the time of writing can now be found within the
 \textsc{Genesis} simulator simulator distribution.\footnote{see \url{http://genesis-sim.org}.}
@@ -86,7 +89,7 @@ With the model boundary conditions,
     \begin{align}
 	v(x, 0) &= E, \\
 	\left.\frac{\partial v}{\partial x}\right\vert_{x=0} & = -Ir, \\
-	\left.\frac{\partial v}{\partial x}\right\vert_{x=L} & = 0,
+	 \left.\frac{\partial v}{\partial x}\right\vert_{x=L} & = 0,
     \end{align}
 \end{subequations}
 where $I$ is the injected current and $L$ is the cable length.
@@ -116,7 +119,11 @@ by
 
 \section{Solution to the normalized cable equation}
 
-Let $G(x, s)$ be the Laplace transform of $g(x, t)$ with respect to $t$. From
+Let $G(x, s)$ be the Laplace transform of $g(x, t)$ with respect to $t$.
+If the inverse transform of $G$ exists, it must agree with $g$ almost everywhere
+for $t>0$, or for all $t>0$ given the continuity of $g$.
+
+From
 \eqref{eq:normcable} and \eqref{eq:normcableinitial},
 \begin{equation}
     \label{eq:lap}
@@ -155,19 +162,162 @@ Consequently,
     \end{aligned}
 \end{equation}
 
-$G(x, s)$ has poles when $s=0$, $m=0$, or when $mL = k\pi i$ for some non-zero
-integer $k$.
+\subsection*{Inversion}
+
+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}):
+\begin{enumerate}
+\item
+    $G(x, s)$ is analytic in some right half-plane $H$,
+\item
+    the singularities of $G(x, s)$ are all poles, and
+\item for some $k>1$, $|s^k G(x, s)|$ is bounded in $H$ and
+    on the circles $|s|=\rho_i$, where $\rho_i$ is an unbounded monotonically
+    increasing sequence.
+\end{enumerate}
+If in addition all the poles $\sigma_j$ of $G$ are simple, then
+the series expansion of the inverse transform is given by
+\begin{equation}
+    g(x,t) = \sum_j e^{\sigma_j t}\, \Res(G;\sigma_j),\quad t>0.
+\end{equation}
+
+$G(x,s)$ has poles when $s=0$, $m=0$, or $\sinh mL=0$.
+In the following, let
+\begin{gather*}
+    a_k = k\pi /L\\
+    m_k = a_k i\\
+    s_k = m_k^2 -1 = -k^2\pi^2/L^2-1.
+\end{gather*}
+for $k\in\mathbb{Z}$.
+The poles are then $0$, $-1$, and $s_k$ for $k\geq 1$.
+
+There are no branch points
+arising from $m=\sqrt{1+s}$, as letting $m=-\sqrt{1+s}$ leaves $G$ unchanged.
+
+For $|s+1|>\epsilon$,
+\begin{equation}
+    \begin{aligned}
+	|s^{3/2}G(x,s)|^2
+	    & \leq (1+\epsilon)^{-1} \left| \frac{\cosh m(L-x)}{\sinh mL} \right|^2
+	\\
+	    & \leq (1+\epsilon)^{-1} (1+|\coth mL|)^2
+    \label{eq:gbounds}
+    \end{aligned}
+\end{equation}
+which is bounded in the half-plane $\Re(s) \geq  -1+\epsilon$. 
+As $\coth u+iv$ is periodic in $v$, \eqref{eq:gbounds} is also bounded outside
+the regions $\{s: m\in D_k\}$, where $D_k$ are non-overlapping $\delta$-neighborhoods of $m_k$
+for some fixed small $\delta>0$. These are contained within disks of radius
+$2k\pi\delta/L+\delta^2$ 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
-a neighbourhood of $s=0$, and so
+\textbf{Pole at $s=0$}.\\
+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}
-	\Res(G, 0) & = - \frac{1}{m}\cdot\left.\frac{\cosh m(L-x)}{\sinh mL}\right|_{s=0}\\
+	\Res(G; 0) & = - \frac{1}{m}\cdot\left.\frac{\cosh m(L-x)}{\sinh mL}\right|_{s=0}\\
 		   & = - \frac{\cosh (L-x)}{\sinh L}.
     \end{aligned}
 \end{equation}
 
-%
+\textbf{Pole at $s=-1$}.\\
+Let $G(x,s)=f(x,s)/h(s)$, where
+\begin{equation}
+    \begin{aligned}
+        f(x, s) &= -\frac{1}{s}\cosh m(L-x)\\
+        h(s) &= m\sinh mL.
+    \end{aligned}
+    \label{eq:hf}
+\end{equation}
+Noting that $dm/ds = \frac{1}{2}m^{-1}$,
+\begin{equation}
+    \begin{aligned}
+	h'(s) &= \frac{1}{2}m^{-1}\sinh mL + \frac{1}{2}L\cosh mL \\
+	      &= \frac{1}{2}L + \frac{1}{2}L + O(m^2) \quad(m\to 0) \\
+	      &= L + O(s+1) \quad(s\to -1).
+    \label{eq:hprime}
+    \end{aligned}
+\end{equation}
+The pole is therefore simple, and
+\begin{equation}
+    \Res(G; -1) = f(x, -1)/h'(-1) = -\frac{1}{L}.
+\end{equation}
+
+\textbf{Pole at $s=s_k$, $k \geq 1$}.\\
+Taking $h$ and $f$ as above \eqref{eq:hf},
+$m_k$ is non-zero for $k\geq 1$ and
+\[
+    h'(s_k) = \frac{1}{2}L\cosh m_kL.
+\]
+Consequently the pole is simple and
+\begin{equation}
+    \begin{aligned}
+	\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}\cosh m_k x,
+    \end{aligned}
+\end{equation}
+as $\sinh m_k=0$.
+
+In terms of $a_k$,
+\begin{equation}
+    \Res(G; s_k) = \frac{2}{L}\cdot\frac{1}{1+a_k^2}\cdot\cos a_kx.
+\end{equation}
+
+The series exapnsion for $g(x, t)$ therefore is
+\begin{equation}
+    g(x, t) = -\frac{\cosh(L-x)}{\sinh L} + \frac{1}{L}e^{-t}\left\{
+	1+2\sum_{k=1}^\infty \frac{e^{-ta_k^2}}{1+a_k^2}\cos a_k x\right\}.
+    \label{eq:theg}
+\end{equation}
+
+\section{Computation of series}
+
+As $t$ approaches zero, the series \eqref{eq:theg} converges increasingly slowly.
+For computational purposes, an estimate on the series residual allows the determination
+of stopping criteria for a given tolerance.
+
+Let $g_n$ be the partial sum
+\begin{equation}
+    g_n(x, t) = -\frac{\cosh(L-x)}{\sinh L} + \frac{1}{L}e^{-t}\left\{
+	1+2\sum_{k=1}^n \frac{e^{-ta_k^2}}{1+a_k^2}\cos a_k x\right\}.
+\end{equation}
+so that $g(x, t) =\lim_{n\to\infty} g_n(x,t)$. Let $\bar{g}_n = |g-g_n|$ be the
+residual. The $a_k$ form an increasing sequence, so
+\begin{equation}
+    \begin{aligned}
+	\bar{g}_n(x,t)
+	    & \leq \frac{2}{L}e^{-t}\sum_{n+1}^\infty\frac{e^{-ta_k^2}}{1+a_k^2}\\
+	    & \leq \frac{2}{L}e^{-t}\int_{a_n}^\infty \frac{e^{-tu^2}}{1+u^2}\,du\\
+	    & < \frac{2}{L}e^{-t}\int_{a_n}^\infty \frac{e^{-tu^2}}{u^2}\,du.
+    \end{aligned}
+    \label{eq:gbar}
+\end{equation}
+Substituting $u=v/\sqrt{t}$ 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}\,\Gamma(-\frac{1}{2},a^2 t),
+\end{equation}
+where $\Gamma(\alpha, z)$ is the upper incomplete gamma function.
+
+For real $\alpha<1$ and $z>0$, \textcite[][Theorem 2.3]{borwein2009} give the upper bound
+\[
+    \Gamma(\alpha, z) \leq z^{\alpha-1} e^{-z}.
+\]
+Substituting into \eqref{eq:gbar} gives
+\begin{equation}
+    \begin{aligned}
+	\bar{g}_n(x,t)
+	    & < \frac{1}{L}e^{-t}\sqrt{t}\,\Gamma(-\frac{1}{2},a_n^2 t) \\
+	    & \leq \frac{1}{L}e^{-t}\sqrt{t}\,(a_n^2 t)^{-\frac{3}{2}}e^{-a_n^2t} \\
+            & = \frac{e^{-t(1+a_n^2)}}{L t a_n^3}.
+    \end{aligned}
+\end{equation}
 
 \printbibliography
 \end{document}

docs/passive_cable/makefile

+SOURCES := cable_computation.tex
+
+LATEXMK := latexmk -e '$$clean_ext=q/bbl run.xml/' -xelatex -use-make -halt-on-error
+
+all: cable_computation.pdf
+
+cable_computation.pdf: cable_computation.tex cable.bib
+	$(LATEXMK) $<
+
+clean:
+	for s in $(SOURCES); do $(LATEXMK) -c "$$s"; done
+
+realclean:
+	for s in $(SOURCES); do $(LATEXMK) -C "$$s"; done

scripts/PassiveCable.jl

@@ -86,7 +86,7 @@ end
 #     computed potential at (x,t) on cable.
 
 function cable(x, t, L, lambda, tau, r, V, I; tol=1e-8)
-    scale = -I*r*lambda;
+    scale = I*r*lambda;
     if scale == 0
         return V
     else
@@ -135,7 +135,7 @@ function rallpack1(x, t; tol=1e-8)
     lambda = sqrt(d/4 * RM/RA)
     tau = CM*RM
 
-    return cable(x, t, L, lambda, tau, r, EM, I, tol=tol)
+    return cable(x, t, L, lambda, tau, r, EM, -I, tol=tol)
 end
 
 end # module PassiveCable