
\chapter{Introduction}

The goal of the scientific treat of any function; and, in particular, of a holomorphic function,
is the conversion to a special function.

Definition.

Function \(f\) is qualified as «special function», if the following conditions are satisfied:

1. The definition of function   \(f\)  is available in the free access.

2. The properties of   function   \(f\)  are investigated and available in the free access; in particular, the asymptotics.

3. Relations with other special functions are revealed.

4. The efficient algorithm of the evaluation is implemented and available in the free access.

Often, the efficient evaluation of a functions is performed through its asymptotic expansions.

In this book, I try to understand, what mathematical object can be qualified as an asymptotic of a function.

\section{Naive approach}

In the simplest approach we do not say anything about range of validity of the asymptotic;
we just say, that the argument should have a huge value or a small value (or the difference between and argument and some constant should be a small value). 

In this approach we assume, that, by default the argument of a function is element of some default set.
Usually, it is set of real numbers or set of complex numbers.
