2 Asymptotic Approximations Areas 2 Asymptotic Approximations 2.2 Transcendental Equations

§2.1 Definitions and Elementary Properties

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Permalink:: http://dlmf.nist.gov/2.1
See also:: Annotations for Ch.2

§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

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Keywords:: asymptotic and order symbols, definition
Notes:: See Olver (1997b, pp. 4–8).
Referenced by:: §34.8
Permalink:: http://dlmf.nist.gov/2.1.i
See also:: Annotations for §2.1 and Ch.2

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

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

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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 Ch.2

2.1.2 $\displaystyle f(x)=o\left(\phi(x)\right)\Longleftrightarrow f(x)/\phi(x)\to 0.$

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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 Ch.2

2.1.3 $\displaystyle f(x)=O\left(\phi(x)\right)\Longleftrightarrow|f(x)/\phi(x)|\text% { is bounded.}$

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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 Ch.2

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

Examples

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See also:: Annotations for §2.1(i), §2.1 and Ch.2

2.1.4 $\tanh x\sim x,$ $x\to 0$ in $\mathbb{C}$ .

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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 Ch.2

2.1.5 $e^{-x}=o\left(1\right),$ $x\to+\infty$ in $\mathbb{R}$ .

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Symbols:: $\mathrm{e}$ : base of natural logarithm, $o\left(\NVar{x}\right)$ : 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 Ch.2

2.1.6 $\sin\left(\pi x+x^{-1}\right)=O\left(x^{-1}\right),$ $x\to\pm\infty$ in $\mathbb{Z}$ .

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathbb{Z}$ : set of all integers and $\sin\NVar{z}$ : sine function
Permalink:: http://dlmf.nist.gov/2.1.E6
Encodings:: TeX, pMML, png
See also:: Annotations for §2.1(i), §2.1(i), §2.1 and Ch.2

2.1.7 $e^{ix}=O\left(1\right),$ $x\in\mathbb{R}$ .

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\in$ : element of, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit 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 Ch.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 and let $t\to\pm\infty$ .)

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

2.1.8 $\sum_{s=n}^{\infty}a_{s}z^{s}=O\left(z^{n}\right),$ $z\to 0$ in $\mathbb{C}$ .

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\mathbb{C}$ : complex plane, $a_{s}$ : coefficients and : 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 Ch.2

Example

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See also:: Annotations for §2.1(i), §2.1 and Ch.2

2.1.9 $e^{z}=1+z+O\left(z^{2}\right),$ $z\to 0$ in $\mathbb{C}$ .

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\mathbb{C}$ : complex plane and $\mathrm{e}$ : base of natural logarithm
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 Ch.2

The symbols and can be used generically. For example,

2.1.10

$o\left(\phi\right)=O\left(\phi\right)$ ,

$o\left(\phi\right)+o\left(\phi\right)=o\left(\phi\right)$ ,

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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 Ch.2

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

§2.1(ii) Integration and Differentiation

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Keywords:: Ritt’s theorem, asymptotic and order symbols, differentiation, differentiation of asymptotic approximations, integration
Notes:: See Olver (1997b, pp. 8–11).
Referenced by:: §2.1(iii), §5.15, (9.12.26), (9.12.28)
Permalink:: http://dlmf.nist.gov/2.1.ii
See also:: Annotations for §2.1 and Ch.2

Integration of asymptotic and order relations is permissible, subject to obvious convergence conditions. For example, suppose 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 $\int_{x}^{\infty}f(t)\,\mathrm{d}t\sim-\frac{x^{\nu+1}}{\nu+1},$ $\Re\nu<-1$ ,

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Symbols:: $\sim$ : asymptotic equality, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\Re$ : real part, : continuous function and $\nu$ : complex constant
Permalink:: http://dlmf.nist.gov/2.1.E11
Encodings:: TeX, pMML, png
See also:: Annotations for §2.1(ii), §2.1 and Ch.2

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

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Symbols:: $\sim$ : asymptotic equality, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $\Re$ : real part, : continuous function and $\nu$ : complex constant
Permalink:: http://dlmf.nist.gov/2.1.E12
Encodings:: TeX, pMML, png
See also:: Annotations for §2.1(ii), §2.1 and Ch.2

Differentiation requires extra conditions. For example, if is analytic for all sufficiently large 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 ’s replaced by ’s.

§2.1(iii) Asymptotic Expansions

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Defines:: $\sim$ : Poincaré asymptotic expansion
Keywords:: Poincaré type, algebraic operations, asymptotic approximations and expansions, differentiation, integration, logarithms of, null, powers of, substitution of, uniqueness
Notes:: See Olver (1997b, pp. 16–22).
Referenced by:: §10.17(ii), §10.19(ii), §10.40(i), §10.41(iv), §10.67(i), §2.1(iv), §5.15, Erratum (V1.0.12) for Subsection 2.1(iii)
Permalink:: http://dlmf.nist.gov/2.1.iii
Addition (effective with 1.0.12):: A short paragraph dealing with asymptotic approximations that are expressed in terms of two or more Poincaré asymptotic expansions has been added below (2.1.16).
See also:: Annotations for §2.1 and Ch.2

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

2.1.13 $f(x)=\sum_{s=0}^{n-1}a_{s}x^{-s}+O\left(x^{-n}\right)$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, : function, : 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 Ch.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 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 $\mathbf{X}$ .

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Symbols:: $\sim$ : Poincaré asymptotic expansion, : 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 Ch.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 $\mathbf{X}$ ,

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Symbols:: : function, $\mathbf{X}$ : unbounded set, : 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 Ch.2

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

If 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 $\mathbf{X}$ ,

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Symbols:: $\sim$ : Poincaré asymptotic expansion, : function, $\mathbf{X}$ : unbounded set, : limit point and $a_{s}$ : coefficients
Referenced by:: §2.1(iii), §2.1(iii), Erratum (V1.0.12) for Subsection 2.1(iii)
Permalink:: http://dlmf.nist.gov/2.1.E16
Encodings:: TeX, pMML, png
See also:: Annotations for §2.1(iii), §2.1 and Ch.2

means that for each , the difference between and the 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 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 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 $0+0\cdot z^{-1}+0\cdot z^{-2}+\cdots,$ $z\to\infty$ .

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Permalink:: http://dlmf.nist.gov/2.1.E17
Encodings:: TeX, pMML, png
See also:: Annotations for §2.1(iii), §2.1 and Ch.2

§2.1(iv) Uniform Asymptotic Expansions

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Keywords:: asymptotic approximations and expansions, uniform
Referenced by:: §2.1(v)
Permalink:: http://dlmf.nist.gov/2.1.iv
See also:: Annotations for §2.1 and Ch.2

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 is a parameter (or set of parameters) ranging over a point set (or sets) $\mathbf{U}$ , and for each nonnegative integer

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

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

2.1.18 $f(u,x)\sim\sum_{s=0}^{\infty}a_{s}(u)x^{-s}$

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Symbols:: $\sim$ : Poincaré asymptotic expansion, : parameter (or set), : 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 Ch.2

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

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

§2.1(v) Generalized Asymptotic Expansions

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Keywords:: asymptotic approximations and expansions, asymptotic scale or sequence, generalized
Notes:: See Olver (1997b, pp. 24–27).
Referenced by:: §10.41(v), §11.6(ii), §14.15(i), §8.11(ii)
Permalink:: http://dlmf.nist.gov/2.1.v
See also:: Annotations for §2.1 and Ch.2

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

2.1.19 $\phi_{s+1}(x)=o\left(\phi_{s}(x)\right),$ $x\to c$ in $\mathbf{X}$ ,

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Symbols:: $o\left(\NVar{x}\right)$ : order less than, $\mathbf{X}$ : point set, : 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 Ch.2

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

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\mathbf{X}$ : point set, : limit point, $\phi_{s}(x)$ : sequence of functions, : function and : nonnegative integer
Permalink:: http://dlmf.nist.gov/2.1.E20
Encodings:: TeX, pMML, png
See also:: Annotations for §2.1(v), §2.1 and Ch.2

for $n=0,1,2,\dots$ . Then $\sum f_{s}(x)$ is a generalized asymptotic expansion of 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 $\mathbf{X}$ .

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Symbols:: $\mathbf{X}$ : point set, : limit point, $\phi_{s}(x)$ : sequence of functions and : function
Permalink:: http://dlmf.nist.gov/2.1.E21
Encodings:: TeX, pMML, png
See also:: Annotations for §2.1(v), §2.1 and Ch.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).

2 Asymptotic Approximations 2.2 Transcendental Equations

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