\chapter{Asymptotic Objects}

\section{Function, Approximation, Residual}

Let $f$ be a function and $A$ be another function.
The difference
\[
r(z)=f(z)-A(z)
\]
is called the residual.

\section{Asymptotic Relation}

We say that $A$ is an asymptotic of $f$ as $z\to\infty$
in a domain $D\subset\mathbb{C}$ if
\[
\lim_{|z|\to\infty,\ z\in D} \frac{r(z)}{f(z)} = 0.
\]

\section{Restricted Asymptotic}

If the limit holds only in a restricted domain $D$,
the asymptotic is called restricted.

\section{Sectorial Asymptotic}

If $D$ can be chosen as a sector
\[
D=\{z\in\mathbb{C} : |\arg z|<\alpha\},
\]
the asymptotic is called sectorial.

\section{Non-uniformity}

The convergence above is not required to be uniform
with respect to the sector parameter $\alpha$.
This non-uniformity is essential near singular directions.

\section{Agreement as a Quantitative Measure}

In numerical applications,
the quality of an asymptotic approximation
can be quantified using agreement functions,
introduced in a later chapter.