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


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



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



Related function: \href{http://algo.inria.fr/esf/function/ERFC/ERFC.html#ERFC}{Complementary Error Function}
\section*{ERF.2 Series and asymptotic expansions}
\label{ERF:asympt}
\subsection*{ERF.2.1 Taylor expansion at $0$}
\label{743360358532860554}
\subsubsection*{ERF.2.1.1 First terms}
\label{743359593218213909}



\begin{equation*}
\label{ERF:asympt:0:RDLBLRDTERMSRDEQ}
\begin{split}
\operatorname{erf} (x)& =\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{ERF.2.1.1.1}
\end{equation*}

\subsubsection*{ERF.2.1.2 General form}
\label{743360426612815574}



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

\subsection*{ERF.2.2 Asymptotic expansion at $\infty$}
\label{743360990165494891}
\subsubsection*{ERF.2.2.1 First terms}
\label{743360340682293628}



\begin{equation*}
\begin{split}
& \operatorname{erf} (x)\approx \operatorname{e} ^{\biggl(-\frac{1}{x^{2}}\biggr)} x y _{0} (x) + ser _{\bigl[1,1,\bigl[\bigl[0,\bigl[[0,1]\bigr]\bigr]\bigr]\bigr]},
\end{split}
\end{equation*}
where
\begin{equation*}
\begin{split}
y _{0} (x)& =-\frac{1}{\sqrt{\pi}} + \frac{x^{2}}{2 \sqrt{\pi}} + 2 \ldots \\
y _{1} (x)& =terms _{\bigl[1,1,\bigl[\bigl[0,\bigl[[0,1]\bigr]\bigr]\bigr]\bigr]} + \ldots
\end{split}
\end{equation*}

\subsubsection*{ERF.2.2.2 General form}
\label{743360821284243570}
\paragraph*{ERF.2.2.2.1 Auxiliary function $y _{0} (x)$}
\label{743360837592361123}
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 (0)& =-\frac{1}{\sqrt{\pi}} \\
u (1)& =0
\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*}
\paragraph*{ERF.2.2.2.2 Auxiliary function $y _{1} (x)$}
\label{743360904147485972}
The auxiliary function $y _{1} (x)$ has the exact form 
\begin{equation*}
\begin{split}
y _{1} (x)& =1
\end{split}
\end{equation*}
\section*{ERF.3 Graphs}
\label{743428848269719318}
\subsection*{ERF.3.1 Real axis}
\label{743428391194454997}
\begin{center}
\includegraphics[width=6cm]{ERF/744157630868014077}
\end{center}
\subsection*{ERF.3.2 Complex plane}
\label{743428979133097279}
\begin{center}
\includegraphics[width=6cm]{ERF/744157545089029180}
\end{center}
\end{document}