§2.1 Definitions and Elementary Properties

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§2.1(i) Asymptotic and Order Symbols
§2.1(ii) Integration and Differentiation
§2.1(iii) Asymptotic Expansions
§2.1(iv) Uniform Asymptotic Expansions
§2.1(v) Generalized Asymptotic Expansions

§2.1(i) Asymptotic and Order Symbols

Notes:: See Olver (1997b, pp. 4–8).
Keywords:: asymptotic and order symbols
Referenced by:: §34.8
Permalink:: http://dlmf.nist.gov/2.1.i

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

2.1.1 $\displaystyle f(x)\sim\phi(x)\Longleftrightarrow f(x)/\phi(x)\to 1.$

Defines:: $\sim$ : asymptotic equality
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2.1.2 $\displaystyle f(x)=\mathop{o\/}\nolimits\!\left(\phi(x)\right)\Longleftrightarrow f(x)/\phi(x)\to 0.$

Defines:: $\mathop{o\/}\nolimits\!\left(x\right)$ : order less than
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2.1.3 $\displaystyle f(x)=\mathop{O\/}\nolimits\!\left(\phi(x)\right)\Longleftrightarrow|f(x)/\phi(x)|\text{ is bounded.}$

Defines:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding
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The symbol $\mathop{O\/}\nolimits$ can also apply to the whole set $\mathbf{X}$ , and not just as $x\to c$ .

¶ Examples

2.1.4 $\mathop{\tanh\/}\nolimits x\sim x,$ $x\to 0$ in $\Complex$ .

Symbols:: $\Complex$ : complex plane (excluding infinity), $\mathop{\tanh\/}\nolimits z$ : hyperbolic tangent function and $\sim$ : asymptotic equality
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2.1.5 $e^{{-x}}=\mathop{o\/}\nolimits\!\left(1\right),$ $x\to+\infty$ in $\Real$ .

Symbols:: $e$ : base of exponential function, $\mathop{o\/}\nolimits\!\left(x\right)$ : order less than and $\Real$ : real line (excluding infinity)
Referenced by:: ¶ ‣ §2.1(i)
Permalink:: http://dlmf.nist.gov/2.1.E5
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2.1.6 $\mathop{\sin\/}\nolimits\!\left(\pi x+x^{{-1}}\right)=\mathop{O\/}\nolimits\!\left(x^{{-1}}\right),$ $x\to\pm\infty$ in $\Integer$ .

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\Integer$ : set of all integers and $\mathop{\sin\/}\nolimits z$ : sine function
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2.1.7 $e^{{ix}}=\mathop{O\/}\nolimits\!\left(1\right),$ $x\in\Real$ .

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\in$ : element of, $e$ : base of exponential function and $\Real$ : real line (excluding infinity)
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In (2.1.5) $\Real$ can be replaced by any fixed ray in the sector $|\mathop{\mathrm{ph}\/}\nolimits x|<\frac{1}{2}\pi$ , or by the whole of the sector $|\mathop{\mathrm{ph}\/}\nolimits x|\leq\frac{1}{2}\pi-\delta$ . (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 $|\mathop{\mathrm{ph}\/}\nolimits x|<\frac{1}{2}\pi$ (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 $\sum _{{s=n}}^{{\infty}}a_{s}z^{s}=\mathop{O\/}\nolimits\!\left(z^{n}\right),$ $z\to 0$ in $\Complex$ .

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\Complex$ : complex plane (excluding infinity), $a_{s}$ : coefficients and $n$ : nonnegative integer
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¶ Example

2.1.9 $e^{z}=1+z+\mathop{O\/}\nolimits\!\left(z^{2}\right),$ $z\to 0$ in $\Complex$ .

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\Complex$ : complex plane (excluding infinity) and $e$ : base of exponential function
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The symbols $\mathop{o\/}\nolimits$ and $\mathop{O\/}\nolimits$ can be used generically. For example,

2.1.10

$\mathop{o\/}\nolimits\!\left(\phi\right)=\mathop{O\/}\nolimits\!\left(\phi\right)$ ,

$\mathop{o\/}\nolimits\!\left(\phi\right)+\mathop{o\/}\nolimits\!\left(\phi\right)=\mathop{o\/}\nolimits\!\left(\phi\right)$ ,

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding and $\mathop{o\/}\nolimits\!\left(x\right)$ : order less than
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it being understood that these equalities are not reversible. (In other words $=$ here really means $\subseteq$ .)

§2.1(ii) Integration and Differentiation

Notes:: See Olver (1997b, pp. 8–11).
Keywords:: Ritt’s theorem, asymptotic and order symbols, differentiation, integration
Referenced by:: §2.1(iii), §5.15, §9.12(viii)
Permalink:: http://dlmf.nist.gov/2.1.ii

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 $\Real$ , where $\nu$ ( $\in\Complex$ ) is a constant. Then

2.1.11 $\int _{{x}}^{{\infty}}f(t)dt\sim-\frac{x^{{\nu+1}}}{\nu+1},$ $\realpart{\nu}<-1$ ,

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : real part, $\sim$ : asymptotic equality, $f(x)$ : continuous function and $\nu$ : complex constant
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2.1.12 $\int f(x)dx\sim\begin{cases}\text{a constant,}&\realpart{\nu}<-1,\\ \mathop{\ln\/}\nolimits x,&\nu=-1,\\ x^{{\nu+1}}/(\nu+1),&\realpart{\nu}>-1.\end{cases}$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\realpart{}$ : real part, $\sim$ : asymptotic equality, $f(x)$ : continuous function and $\nu$ : complex constant
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Differentiation requires extra conditions. For example, if $f(z)$ is analytic for all sufficiently large $|z|$ in a sector $\mathbf{S}$ and $f(z)=\mathop{O\/}\nolimits\!\left(z^{{\nu}}\right)$ as $z\to\infty$ in $\mathbf{S}$ , $\nu$ being real, then $f^{{\prime}}(z)=\mathop{O\/}\nolimits\!\left(z^{{\nu-1}}\right)$ 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 $\mathop{O\/}\nolimits$ ’s replaced by $\mathop{o\/}\nolimits$ ’s.

§2.1(iii) Asymptotic Expansions

Notes:: See Olver (1997b, pp. 16–22).
Keywords:: asymptotic approximations and expansions, differentiation, integration
Referenced by:: §10.17(ii), §10.19(ii), §10.40(i), §10.41(iv), §10.67(i), §2.1(iv), §5.15
Permalink:: http://dlmf.nist.gov/2.1.iii

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

2.1.13 $f(x)=\sum _{{s=0}}^{{n-1}}a_{s}x^{{-s}}+\mathop{O\/}\nolimits\!\left(x^{{-n}}\right)$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $f(x)$ : function, $n$ : nonnegative integer and $a_{s}$ : coefficients
Referenced by:: §2.1(iii), §2.1(iv)
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as $x\to\infty$ in an unbounded set $\mathbf{X}$ in $\Real$ or $\Complex$ . 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:: $\sim$ : asymptotic equality, $f(x)$ : function, $\mathbf{X}$ : unbounded set and $a_{s}$ : coefficients
Referenced by:: §2.1(iii)
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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
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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 $\Complex$ .

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:: $\sim$ : asymptotic equality, $f(x)$ : function, $\mathbf{X}$ : unbounded set, $c$ : limit point and $a_{s}$ : coefficients
Referenced by:: §2.1(iii)
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means that for each $n$ , the difference between $f(x)$ and the $n$ th partial sum on the right-hand side is $\mathop{O\/}\nolimits\!\left((x-c)^{n}\right)$ 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 $\mathop{O\/}\nolimits$ symbol (§2.1(ii)). For reversion see §2.2.

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 $\Complex$ of finite angle. As an example, in the sector $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\frac{1}{2}\pi-\delta$ ( $<\frac{1}{2}\pi$ ) each of the functions $0,e^{{-z}}$ , and $e^{{-\sqrt{z}}}$ (principal value) has the null asymptotic expansion

2.1.17 $0+0\cdot z^{{-1}}+0\cdot z^{{-2}}+\cdots,$ $z\to\infty$ .

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§2.1(iv) Uniform Asymptotic Expansions

Keywords:: asymptotic approximations and expansions
Referenced by:: §2.1(v)
Permalink:: http://dlmf.nist.gov/2.1.iv

If the set $\mathbf{X}$ in §2.1(iii) is a closed sector $\alpha\leq\mathop{\mathrm{ph}\/}\nolimits x\leq\beta$ , then by definition the asymptotic property (2.1.13) holds uniformly with respect to $\mathop{\mathrm{ph}\/}\nolimits x\in[\alpha,\beta]$ 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:: $\sim$ : asymptotic equality, $u$ : parameter (or set), $f(u,x)$ : function and $a_{s}(u)$ : coefficients
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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

Notes:: See Olver (1997b, pp. 24–27).
Defines:: $\{\ldots\}$ : sequence, asymptotic sequence (or scale), or enumerable set
Keywords:: asymptotic approximations and expansions, asymptotic scale or sequence
Referenced by:: §10.41(v), §11.6(ii), §14.15(i), §8.11(ii)
Permalink:: http://dlmf.nist.gov/2.1.v

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 $\phi _{{s+1}}(x)=\mathop{o\/}\nolimits\!\left(\phi _{s}(x)\right),$ $x\to c$ in $\mathbf{X}$ ,

Symbols:: $\mathop{o\/}\nolimits\!\left(x\right)$ : order less than, $\mathbf{X}$ : point set, $c$ : limit point and $\phi _{s}(x)$ : sequence of functions
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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 $f(x)=\sum _{{s=0}}^{{n-1}}f_{s}(x)+\mathop{O\/}\nolimits\!\left(\phi _{n}(x)\right),$ $x\to c$ in $\mathbf{X}$ ,

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathbf{X}$ : point set, $c$ : limit point, $\phi _{s}(x)$ : sequence of functions, $f(x)$ : function and $n$ : nonnegative integer
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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:: $\{\ldots\}$ : sequence, asymptotic sequence (or scale), or enumerable set, $\sim$ : asymptotic equality, $\mathbf{X}$ : point set, $c$ : limit point, $\phi _{s}(x)$ : sequence of functions and $f(x)$ : function
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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).