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\begin{document}
\chapter*{ERFC Complementary Error Function}
\label{ERFC}
\section*{ERFC.1 Introduction}
\label{ERFC:intro}


Let $x$ be a complex variable of $\mathbb{C} \setminus \{\infty\}$.The function Complementary Error Function (noted $\operatorname{erfc}$) is defined by the following second order differential equation
\begin{equation*}
\label{ERFC:diffeq}
\begin{split}
2 x \frac{\partial y (x)}{\partial x} + \frac{\partial^{2} y (x)}{\partial x^{2}}& =0.
\end{split}\tag{ERFC.1.1}
\end{equation*}



The initial conditions of ERFC.1.1 are given at $0$ by
\begin{equation*}
\label{ERFC:inicond}
\begin{split}
\operatorname{erfc} (0)& =1, \\
\frac{\partial \operatorname{erfc} (x)}{\partial x} (0)& =\frac{-2}{\sqrt{\pi}}.
\end{split}\tag{ERFC.1.2}
\end{equation*}



Related function: \href{http://algo.inria.fr/esf/function/ERF/ERF.html#ERF}{Error Function}
\section*{ERFC.2 Series and asymptotic expansions}
\label{ERFC:asympt}
\subsection*{ERFC.2.1 Taylor expansion at $0$}
\label{743429268662169964}
\subsubsection*{ERFC.2.1.1 First terms}
\label{74342878532217691}



\begin{equation*}
\label{ERFC:asympt:0:RDLBLRDTERMSRDEQ}
\begin{split}
\operatorname{erfc} (x)& =1 - \frac{2}{\sqrt{\pi}}   x + \frac{2}{3 \sqrt{\pi}}   x^{3} - \frac{1}{5 \sqrt{\pi}}   x^{5} + \frac{1}{21 \sqrt{\pi}}   x^{7} - \frac{1}{108 \sqrt{\pi}}   x^{9} +  \\
& \quad{}\quad{}\frac{1}{660 \sqrt{\pi}}   x^{11} - \frac{1}{4680 \sqrt{\pi}}   x^{13} + \frac{1}{37800 \sqrt{\pi}}   x^{15} + \operatorname{O} \bigl(x^{16}\bigr).
\end{split}\tag{ERFC.2.1.1.1}
\end{equation*}

\subsubsection*{ERFC.2.1.2 General form}
\label{743429464104997179}



\begin{equation*}
\label{ERFC:asympt:0:RDLBLRDGENFORMRDEQ}
\begin{split}
\operatorname{erfc} (x)& =\sum_{n = 0}^{\infty} u (n) x^{n}.
\end{split}\tag{ERFC.2.1.2.1}
\end{equation*}
The coefficients $u (n)$ satisfy the recurrence
\begin{equation*}
\label{ERFC:asympt:0:toto}
\begin{split}
2 n u (n) + \bigl(n^{2} + 3 n + 2\bigr) u (n + 2)& =0.
\end{split}\tag{ERFC.2.1.2.2}
\end{equation*}
Initial conditions of ERFC.2.1.2.2 are given by
\begin{equation*}
\label{ERFC:asympt:0:RDLBLRDGENFORMRDIC}
\begin{split}
u (1)& =\frac{-2}{\sqrt{\pi}}, \\
u (0)& =1.
\end{split}\tag{ERFC.2.1.2.3}
\end{equation*}

\subsection*{ERFC.2.2 Asymptotic expansion at $\infty$}
\label{743429795452072871}
\subsubsection*{ERFC.2.2.1 First terms}
\label{743429859404136675}



\begin{equation*}
\begin{split}
& \operatorname{erfc} (x)\approx \operatorname{e} ^{\biggl(-\frac{1}{x^{2}}\biggr)} x y _{0} (x),
\end{split}
\end{equation*}
where
\begin{equation*}
\begin{split}
y _{0} (x)& =\pi^{\Bigl(-\frac{1}{2}\Bigr)} - \frac{x^{2}}{2 \sqrt{\pi}} + 2 \ldots
\end{split}
\end{equation*}

\subsubsection*{ERFC.2.2.2 General form}
\label{743429850685116669}
\paragraph*{ERFC.2.2.2.1 Auxiliary function $y _{0} (x)$}
\label{743429511584194463}
The coefficients $u (n)$ of $y _{0} (x)$ satisfy the following recurrence
\begin{equation*}
\begin{split}
2 n u (n) + u (n - 2) \bigl(-4 + 3 n + (n - 2)^{2}\bigr)& =0
\end{split}
\end{equation*}
whose initial conditions are given by
\begin{equation*}
\begin{split}
u (1)& =0 \\
u (0)& =\pi^{\Bigl(-\frac{1}{2}\Bigr)}
\end{split}
\end{equation*}
This recurrence has the closed form solution
\begin{equation*}
\begin{split}
u (2 n + 1)& =0, \\
u (2 n)& =\frac{(-1)^{n} \Gamma \Bigl(n + \frac{1}{2}\Bigr)}{\pi}.
\end{split}
\end{equation*}
\section*{ERFC.3 Graphs}
\label{743498318537422689}
\subsection*{ERFC.3.1 Real axis}
\label{74349858228080911}
\begin{center}
\includegraphics[width=6cm]{ERFC/74417882790650317}
\end{center}
\subsection*{ERFC.3.2 Complex plane}
\label{743498475079110605}
\begin{center}
\includegraphics[width=6cm]{ERFC/744178882177624648}
\end{center}
\end{document}