= Asymptotic expansion
{title2=$f(z)\sim\sum_{k\geq0}a_k\phi_k(z)$}
{wiki}
= Poincaré expansion
{c}
{synonym}
= Asymptotic series
{synonym}
An <asymptotic expansion> describes a <function> near a specified <limit> by successively smaller scales. Each fixed finite truncation has a remainder smaller than its last retained scale. The central question is how the approximation improves as the parameter approaches the <limit>; convergence of the entire infinite <series> at a fixed parameter is a separate question. For the inverse-power scale $1,z^{-1},z^{-2},\ldots$,
$$
f(z)\sim\sum_{k=0}^{\infty}a_kz^{-k}\quad(z\to+\infty)
$$
means that, for every fixed <nonnegative integer> $m$,
$$
f(z)-\sum_{k=0}^ma_kz^{-k}=o(z^{-m}).
$$
The <little o notation> makes the accuracy requirement precise. With expansions to all orders, this is also equivalent to an $O(z^{-m-1})$ remainder after the $m$th term; the implied bound may depend on $m$.<NIST Digital Library of Mathematical Functions/2.1 Definitions and Elementary Properties>[NIST Digital Library of Mathematical Functions, §2.1(iii), equations 2.1.13–2.1.15: fixed-order inverse-power expansions]{ref}
For a quick convergent example, the finite <geometric series> identity gives
$$
\frac1{1+z^{-1}}
=1-\frac1z+\frac1{z^2}-\cdots+\frac{(-1)^m}{z^m}
+\frac{(-1)^{m+1}z^{-m-1}}{1+z^{-1}}.
$$
For $z>1$, both the <asymptotic expansion> as $z\to+\infty$ and the ordinary convergent <geometric series> describe this <function>. The exact remainder displays the required extra inverse power.
The more surprising example under <a divergent factorial series can approximate a convergent integral> has a convergent <integral> but a divergent <series> at every fixed positive $z$:
$$
F(z)=\int_0^\infty\frac{e^{-t}}{1+t/z}\,dt
\sim1-\frac1z+\frac{2!}{z^2}-\frac{3!}{z^3}+\cdots.
$$
A few terms can estimate $F(10)$ well, even though adding sufficiently many terms eventually destroys the approximation. <Optimal truncation> explains where to stop.
This makes <asymptotic expansions> useful when an exact expression is a complicated <integral> or <differential equation> solution, but a parameter is very large or very small. The leading term reveals the scale; corrections reveal which effects matter next. The <asymptotic sequence> must be specified: integer powers, fractional powers, <logarithms>, and <exponential functions> can encode different kinds of behavior. A good finite approximation includes a remainder estimate or an independently checked range of use.
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