\chapter{Asymptotic Definitions}

This chapter introduces definitions related to asymptotic behavior.
The emphasis is on explicit specification of the domain
in which the asymptotic relations are considered.

\section{Asymptotic Approximation}

Let \(f\) and \(A\) be functions defined on a set \(D\),
and let \(z_0\) be a limit point of \(D\).

The expression
\[
f(z) \sim A(z) \quad \text{as } z \to z_0 \text{ in } D
\]
means that
\[
\lim_{\substack{z \to z_0 \\ z \in D}}
\frac{f(z)}{A(z)} = 1.
\]

The specification of the set \(D\) is an essential part
of the asymptotic statement.

\section{Restricted Asymptotics}

An asymptotic relation is called restricted
if the limit \(z \to z_0\) is taken
only within a proper subset \(D\)
of the natural domain of the functions involved.

Different choices of \(D\)
may lead to different asymptotic interpretations
of the same functions.

\section{Sectorial Asymptotics}

Let \(z_0 = \infty\).
A sectorial asymptotic is an asymptotic relation
in which the set \(D\) is a sector
\[
D = \{ z \in \mathbb{C} : |z| > R,\ \alpha < \arg z < \beta \}.
\]

Sectorial asymptotics are typical
for holomorphic functions
with branch points or Stokes lines.

\section{Asymptotic Expansion}

An asymptotic expansion of a function \(f\)
is a formal series
\[
A(z) = \sum_{k=0}^{\infty} a_k \phi_k(z)
\]
such that for each \(n\),
\[
f(z) - \sum_{k=0}^{n-1} a_k \phi_k(z)
= o\bigl(\phi_{n}(z)\bigr)
\quad \text{as } z \to z_0 \text{ in } D.
\]

The functions \(\phi_k\) are ordered
by decreasing magnitude in the limit considered.

\section{Uniformity}

An asymptotic relation is called uniform
with respect to a parameter
if the convergence in the asymptotic limit
is uniform in that parameter.

Non-uniform asymptotics often require
restriction of the domain
or the use of different asymptotic representations
in different regions.

\section{Remarks}

\begin{itemize}
\item An asymptotic approximation is not required
      to converge.

\item An asymptotic expansion may provide
      highly accurate numerical evaluations
      when truncated appropriately.

\item The same function may admit
      different asymptotic representations
      in different domains.
\end{itemize}
