# §2.1 Definitions and Elementary Properties

## §2.1(i) Asymptotic and Order Symbols

Let $\mathbf{X}$ be a point set with a limit point $c$. As $x\to c$ in $\mathbf{X}$

 2.1.1 $\displaystyle f(x)\sim\phi(x)$ $\displaystyle\Longleftrightarrow f(x)/\phi(x)\to 1.$ ⓘ Defines: $\sim$: asymptotic equality Permalink: http://dlmf.nist.gov/2.1.E1 Encodings: TeX, pMML, png See also: Annotations for 2.1(i), 2.1 and 2 2.1.2 $\displaystyle f(x)=o\left(\phi(x)\right)$ $\displaystyle\Longleftrightarrow f(x)/\phi(x)\to 0.$ ⓘ Defines: $o\left(\NVar{x}\right)$: order less than Permalink: http://dlmf.nist.gov/2.1.E2 Encodings: TeX, pMML, png See also: Annotations for 2.1(i), 2.1 and 2 2.1.3 $\displaystyle f(x)=O\left(\phi(x)\right)$ $\displaystyle\Longleftrightarrow|f(x)/\phi(x)|\text{ is bounded.}$ ⓘ Defines: $O\left(\NVar{x}\right)$: order not exceeding Permalink: http://dlmf.nist.gov/2.1.E3 Encodings: TeX, pMML, png See also: Annotations for 2.1(i), 2.1 and 2

The symbol $O$ can also apply to the whole set $\mathbf{X}$, and not just as $x\to c$.

### Examples

 2.1.4 $\tanh x\sim x,$ $x\to 0$ in $\mathbb{C}$. ⓘ Symbols: $\sim$: asymptotic equality, $\mathbb{C}$: complex plane and $\tanh\NVar{z}$: hyperbolic tangent function Permalink: http://dlmf.nist.gov/2.1.E4 Encodings: TeX, pMML, png See also: Annotations for 2.1(i), 2.1(i), 2.1 and 2
 2.1.5 $e^{-x}=o\left(1\right),$ $x\to+\infty$ in $\mathbb{R}$. ⓘ Symbols: $\mathrm{e}$: base of exponential function, $o\left(\NVar{x}\right)$: order less than and $\mathbb{R}$: real line Referenced by: §2.1(i) Permalink: http://dlmf.nist.gov/2.1.E5 Encodings: TeX, pMML, png See also: Annotations for 2.1(i), 2.1(i), 2.1 and 2
 2.1.6 $\sin\left(\pi x+x^{-1}\right)=O\left(x^{-1}\right),$ $x\to\pm\infty$ in $\mathbb{Z}$.
 2.1.7 $e^{ix}=O\left(1\right),$ $x\in\mathbb{R}$. ⓘ Symbols: $O\left(\NVar{x}\right)$: order not exceeding, $\in$: element of, $\mathrm{e}$: base of exponential function and $\mathbb{R}$: real line Permalink: http://dlmf.nist.gov/2.1.E7 Encodings: TeX, pMML, png See also: Annotations for 2.1(i), 2.1(i), 2.1 and 2

In (2.1.5) $\mathbb{R}$ can be replaced by any fixed ray in the sector $|\operatorname{ph}x|<\frac{1}{2}\pi$, or by the whole of the sector $|\operatorname{ph}x|\leq\frac{1}{2}\pi-\delta$. (Here and elsewhere in this chapter $\delta$ is an arbitrary small positive constant.) But (2.1.5) does not hold as $x\to\infty$ in $|\operatorname{ph}x|<\frac{1}{2}\pi$ (for example, set $x=1+it$ and let $t\to\pm\infty$.)

If $\sum_{s=0}^{\infty}a_{s}z^{s}$ converges for all sufficiently small $|z|$, then for each nonnegative integer $n$

 2.1.8 $\sum_{s=n}^{\infty}a_{s}z^{s}=O\left(z^{n}\right),$ $z\to 0$ in $\mathbb{C}$. ⓘ Symbols: $O\left(\NVar{x}\right)$: order not exceeding, $\mathbb{C}$: complex plane, $a_{s}$: coefficients and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/2.1.E8 Encodings: TeX, pMML, png See also: Annotations for 2.1(i), 2.1(i), 2.1 and 2

### Example

 2.1.9 $e^{z}=1+z+O\left(z^{2}\right),$ $z\to 0$ in $\mathbb{C}$. ⓘ Symbols: $O\left(\NVar{x}\right)$: order not exceeding, $\mathbb{C}$: complex plane and $\mathrm{e}$: base of exponential function Permalink: http://dlmf.nist.gov/2.1.E9 Encodings: TeX, pMML, png See also: Annotations for 2.1(i), 2.1(i), 2.1 and 2

The symbols $o$ and $O$ can be used generically. For example,

 2.1.10 $\displaystyle o\left(\phi\right)$ $\displaystyle=O\left(\phi\right)$, $\displaystyle o\left(\phi\right)+o\left(\phi\right)$ $\displaystyle=o\left(\phi\right)$, ⓘ Symbols: $O\left(\NVar{x}\right)$: order not exceeding and $o\left(\NVar{x}\right)$: order less than Permalink: http://dlmf.nist.gov/2.1.E10 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 2.1(i), 2.1(i), 2.1 and 2

it being understood that these equalities are not reversible. (In other words $=$ here really means $\subseteq$.)

## §2.1(ii) Integration and Differentiation

Integration of asymptotic and order relations is permissible, subject to obvious convergence conditions. For example, suppose $f(x)$ is continuous and $f(x)\sim x^{\nu}$ as $x\to+\infty$ in $\mathbb{R}$, where $\nu$ ($\in\mathbb{C}$) is a constant. Then

 2.1.11 $\displaystyle\int_{x}^{\infty}f(t)\mathrm{d}t$ $\displaystyle\sim-\frac{x^{\nu+1}}{\nu+1},$ $\Re\nu<-1$, 2.1.12 $\displaystyle\int f(x)\mathrm{d}x$ $\displaystyle\sim\begin{cases}\text{a constant,}&\Re\nu<-1,\\ \ln x,&\phantom{\Re}\nu=-1,\\ x^{\nu+1}/(\nu+1),&\Re\nu>-1.\end{cases}$

Differentiation requires extra conditions. For example, if $f(z)$ is analytic for all sufficiently large $|z|$ in a sector $\mathbf{S}$ and $f(z)=O\left(z^{\nu}\right)$ as $z\to\infty$ in $\mathbf{S}$, $\nu$ being real, then $f^{\prime}(z)=O\left(z^{\nu-1}\right)$ as $z\to\infty$ in any closed sector properly interior to $\mathbf{S}$ and with the same vertex (Ritt’s theorem). This result also holds with both $O$’s replaced by $o$’s.

## §2.1(iii) Asymptotic Expansions

Let $\sum a_{s}x^{-s}$ be a formal power series (convergent or divergent) and for each positive integer $n$,

 2.1.13 $f(x)=\sum_{s=0}^{n-1}a_{s}x^{-s}+O\left(x^{-n}\right)$ ⓘ Symbols: $O\left(\NVar{x}\right)$: order not exceeding, $f(x)$: function, $n$: nonnegative integer and $a_{s}$: coefficients Referenced by: §2.1(iii), §2.1(iv) Permalink: http://dlmf.nist.gov/2.1.E13 Encodings: TeX, pMML, png See also: Annotations for 2.1(iii), 2.1 and 2

as $x\to\infty$ in an unbounded set $\mathbf{X}$ in $\mathbb{R}$ or $\mathbb{C}$. Then $\sum a_{s}x^{-s}$ is a Poincaré asymptotic expansion, or simply asymptotic expansion, of $f(x)$ as $x\to\infty$ in $\mathbf{X}$. Symbolically,

 2.1.14 $f(x)\sim a_{0}+a_{1}x^{-1}+a_{2}x^{-2}+\cdots,$ $x\to\infty$ in $\mathbf{X}$. ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $f(x)$: function, $\mathbf{X}$: unbounded set and $a_{s}$: coefficients Referenced by: §2.1(iii) Permalink: http://dlmf.nist.gov/2.1.E14 Encodings: TeX, pMML, png See also: Annotations for 2.1(iii), 2.1 and 2

Condition (2.1.13) is equivalent to

 2.1.15 $x^{n}\left(f(x)-\sum_{s=0}^{n-1}a_{s}x^{-s}\right)\to a_{n},$ $x\to\infty$ in $\mathbf{X}$, ⓘ Symbols: $f(x)$: function, $\mathbf{X}$: unbounded set, $n$: nonnegative integer and $a_{s}$: coefficients Permalink: http://dlmf.nist.gov/2.1.E15 Encodings: TeX, pMML, png See also: Annotations for 2.1(iii), 2.1 and 2

for each $n=0,1,2,\dots$. If $\sum a_{s}x^{-s}$ converges for all sufficiently large $|x|$, then it is automatically the asymptotic expansion of its sum as $x\to\infty$ in $\mathbb{C}$.

If $c$ is a finite limit point of $\mathbf{X}$, then

 2.1.16 $f(x)\sim a_{0}+a_{1}(x-c)+a_{2}(x-c)^{2}+\cdots,$ $x\to c$ in $\mathbf{X}$, ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $f(x)$: function, $\mathbf{X}$: unbounded set, $c$: limit point and $a_{s}$: coefficients Referenced by: §2.1(iii), §2.1(iii), Other Changes Permalink: http://dlmf.nist.gov/2.1.E16 Encodings: TeX, pMML, png See also: Annotations for 2.1(iii), 2.1 and 2

means that for each $n$, the difference between $f(x)$ and the $n$th partial sum on the right-hand side is $O\left((x-c)^{n}\right)$ as $x\to c$ in $\mathbf{X}$.

Most operations on asymptotic expansions can be carried out in exactly the same manner as for convergent power series. These include addition, subtraction, multiplication, and division. Substitution, logarithms, and powers are also permissible; compare Olver (1997b, pp. 19–22). Differentiation, however, requires the kind of extra conditions needed for the $O$ symbol (§2.1(ii)). For reversion see §2.2.

Some asymptotic approximations are expressed in terms of two or more Poincaré asymptotic expansions. In those cases it is usually necessary to interpret each infinite series separately in the manner described above; that is, it is not always possible to reinterpret the asymptotic approximation as a single asymptotic expansion. For an example see (2.8.15).

Asymptotic expansions of the forms (2.1.14), (2.1.16) are unique. But for any given set of coefficients $a_{0},a_{1},a_{2},\dots$, and suitably restricted $\mathbf{X}$ there is an infinity of analytic functions $f(x)$ such that (2.1.14) and (2.1.16) apply. For (2.1.14) $\mathbf{X}$ can be the positive real axis or any unbounded sector in $\mathbb{C}$ of finite angle. As an example, in the sector $|\operatorname{ph}z|\leq\frac{1}{2}\pi-\delta$ ($<\frac{1}{2}\pi$) each of the functions $0,e^{-z}$, and $e^{-\sqrt{z}}$ (principal value) has the null asymptotic expansion

 2.1.17 $0+0\cdot z^{-1}+0\cdot z^{-2}+\cdots,$ $z\to\infty$. ⓘ Permalink: http://dlmf.nist.gov/2.1.E17 Encodings: TeX, pMML, png See also: Annotations for 2.1(iii), 2.1 and 2

## §2.1(iv) Uniform Asymptotic Expansions

If the set $\mathbf{X}$ in §2.1(iii) is a closed sector $\alpha\leq\operatorname{ph}x\leq\beta$, then by definition the asymptotic property (2.1.13) holds uniformly with respect to $\operatorname{ph}x\in[\alpha,\beta]$ as $|x|\to\infty$. The asymptotic property may also hold uniformly with respect to parameters. Suppose $u$ is a parameter (or set of parameters) ranging over a point set (or sets) $\mathbf{U}$, and for each nonnegative integer $n$

 $\left|x^{n}\left(f(u,x)-\sum_{s=0}^{n-1}a_{s}(u)x^{-s}\right)\right|$

is bounded as $x\to\infty$ in $\mathbf{X}$, uniformly for $u\in\mathbf{U}$. (The coefficients $a_{s}(u)$ may now depend on $u$.) Then

 2.1.18 $f(u,x)\sim\sum_{s=0}^{\infty}a_{s}(u)x^{-s}$ ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $u$: parameter (or set), $f(u,x)$: function and $a_{s}(u)$: coefficients Permalink: http://dlmf.nist.gov/2.1.E18 Encodings: TeX, pMML, png See also: Annotations for 2.1(iv), 2.1 and 2

as $x\to\infty$ in $\mathbf{X}$, uniformly with respect to $u\in\mathbf{U}$.

Similarly for finite limit point $c$ in place of $\infty$.

## §2.1(v) Generalized Asymptotic Expansions

Let $\phi_{s}(x)$, $s=0,1,2,\dots$, be a sequence of functions defined in $\mathbf{X}$ such that for each $s$

 2.1.19 $\phi_{s+1}(x)=o\left(\phi_{s}(x)\right),$ $x\to c$ in $\mathbf{X}$, ⓘ Symbols: $o\left(\NVar{x}\right)$: order less than, $\mathbf{X}$: point set, $c$: limit point and $\phi_{s}(x)$: sequence of functions Permalink: http://dlmf.nist.gov/2.1.E19 Encodings: TeX, pMML, png See also: Annotations for 2.1(v), 2.1 and 2

where $c$ is a finite, or infinite, limit point of $\mathbf{X}$. Then $\{\phi_{s}(x)\}$ is an asymptotic sequence or scale. Suppose also that $f(x)$ and $f_{s}(x)$ satisfy

 2.1.20 $f(x)=\sum_{s=0}^{n-1}f_{s}(x)+O\left(\phi_{n}(x)\right),$ $x\to c$ in $\mathbf{X}$,

for $n=0,1,2,\dots$. Then $\sum f_{s}(x)$ is a generalized asymptotic expansion of $f(x)$ with respect to the scale $\{\phi_{s}(x)\}$. Symbolically,

 2.1.21 $f(x)\sim\sum_{s=0}^{\infty}f_{s}(x);\;\;\{\phi_{s}(x)\},$ $x\to c$ in $\mathbf{X}$. ⓘ Symbols: $\mathbf{X}$: point set, $c$: limit point, $\phi_{s}(x)$: sequence of functions and $f(x)$: function Permalink: http://dlmf.nist.gov/2.1.E21 Encodings: TeX, pMML, png See also: Annotations for 2.1(v), 2.1 and 2

As in §2.1(iv), generalized asymptotic expansions can also have uniformity properties with respect to parameters. For an example see §14.15(i).

Care is needed in understanding and manipulating generalized asymptotic expansions. Many properties enjoyed by Poincaré expansions (for example, multiplication) do not always carry over. It can even happen that a generalized asymptotic expansion converges, but its sum is not the function being represented asymptotically; for an example see §18.15(iii).