\chapter{Introduction}

The goal of the scientific treatment of a function,
and in particular of a holomorphic function,
is its conversion into a special function.

In this book, the term ``special function''
is used in an operational sense.

\section{Special Functions}

A function \(f\) is qualified as a special function
if the following conditions are satisfied:

\begin{enumerate}
\item The definition of \(f\) is available in free access.

\item The properties of \(f\) are investigated and documented
      in free access; in particular, its asymptotic behavior
      is described.

\item Relations of \(f\) with other special functions
      are known.

\item An efficient algorithm for the evaluation of \(f\)
      is implemented and available in free access.
\end{enumerate}

The efficient evaluation of special functions
is often achieved through asymptotic approximations.

\section{Evaluation and Approximation}

In this book, a clear distinction is made between
evaluation and approximation.

Evaluation is understood as an action or procedure
that produces a numerical value of a quantity.
The result of an evaluation may be exact or approximate.

An approximation is a function or expression
intended to be close to another function
within a specified domain.

An asymptotic is a special type of approximation
whose quality improves in a limiting regime
and whose validity depends on the domain
in which the limit is taken.

\section{Role of the Domain}

The asymptotic behavior of a function
cannot be characterized independently
of the domain of its input variable.

Different domains may lead to different asymptotic
interpretations of the same function.

This is particularly important for functions
of a complex variable,
where sectorial and restricted asymptotics
naturally arise.

\section{Motivation}

In many practical problems,
the terms ``asymptotic'', ``approximation'',
``estimate'', and ``evaluation''
are used informally and sometimes interchangeably.

Such usage may be sufficient in heuristic discussions,
but it becomes problematic in the context
of numerical implementation and rigorous analysis.

The purpose of this book is to introduce
a consistent terminology and to illustrate it
by explicit examples,
with particular emphasis on holomorphic functions
and their asymptotic behavior.