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\usepackage{graphicx}
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\title{A report on error function}
\author{DVF and the authors of wikipedia}


\begin{abstract}
The report introduces the error function (also called the Gauss error
function) which is a special function (non-elementary) of sigmoid shape which
occurs in probability, statistics and partial differential equations.
\end{abstract}

\maketitle

\section{Introduction}
The error function is defined as~\cite{andrews}

	\begin{equation}\label{eq1}
\mathrm{erf}(x) = \frac{2}{\sqrt{\pi}}\int_{0}^x e^{-t^2}dt \,.
	\end{equation}
The error function is illustrated on Fig.~\ref{plot}.
	\begin{figure}[b]
\centerline{\includegraphics{plot.eps}}
\caption{The plot of the error function. The points indicate the values
obtained by the {\tt erf} function from {\tt stdlib}.}
\label{plot}
	\end{figure}
The complementary error function, denoted erfc, is defined as
	\begin{equation}\label{erfc}
\mathrm{erfc}(x)=1-\mathrm{erf}(x)
=\frac{2}{\sqrt{\pi}}\int_{x}^\infty e^{-t^2}dt \,.
	\end{equation}



\section{The name "error function"}
The error function is used in measurement theory (using probability and
statistics), and although its use in other branches of mathematics has
nothing to do with the characterization of measurement errors, the name has
stuck.
The error function is used in measurement theory (using probability and
statistics), and although its use in other branches of mathematics has
nothing to do with the characterization of measurement errors, the name has
stuck.
The error function is related to the cumulative distribution $\Phi$, the
integral of the standard normal distribution, by~\cite{andrews}.

	\begin{equation}
    \Phi(x)=\frac12+\frac12\mathrm{erf}(\frac{x}{\sqrt{2}}) \;.
	\end{equation}

The error function, evaluated at $\frac{x}{\sigma\sqrt{2}}$ for positive
x values, gives the probability that a measurement, under the influence
of normally distributed errors with standard deviation σ, has a distance
less than x from the mean value.[5] This function is used in statistics to
predict behavior of any sample with respect to the population mean. This
usage is similar to the Q-function, which in fact can be written in
terms of the error function.

\section{Implementation}

The error function is implemented via the complementory error
function, equation~(\ref{erfc}). For positive $x$ the integral in
equation~(\ref{erfc}) is evaluated directly by a call to the {\tt
gsl\_integration\_qagiv} function. For negative $x$ the formula
	\begin{equation}
\mathrm{erfc}(-x)=2-\mathrm{erfc}(x)
	\end{equation}
was used instead.


	\begin{figure}[b]
\centerline{\input{plot2.tex}}
\caption{The plot of the complimentary error function}
\label{plot2}
	\end{figure}

\begin{thebibliography}{9}
\bibitem{andrews}Andrews, Larry C.; Special functions of mathematics for
engineers.
\end{thebibliography}


\end{document}
