2 Asymptotic Approximations Areas 2.1 Definitions and Elementary Properties 2.3 Integrals of a Real Variable

§2.2 Transcendental Equations

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Keywords:: asymptotic solutions, asymptotic solutions of transcendental equations, transcendental equations
Notes:: See Olver (1997b, pp. 11–16) and Fabijonas and Olver (1999).
Referenced by:: §10.70, §2.1(iii)
Permalink:: http://dlmf.nist.gov/2.2
See also:: Annotations for Ch.2

Let $f(x)$ be continuous and strictly increasing when $a<x<\infty$ and

2.2.1		$f(x)\sim x,$
$x\to\infty$ .
ⓘ Symbols: $\sim$ : asymptotic equality and $f(x)$ : function Referenced by: §2.2 Permalink: http://dlmf.nist.gov/2.2.E1 Encodings: TeX, pMML, png See also: Annotations for §2.2 and Ch.2

Then for $y>f(a)$ the equation $f(x)=y$ has a unique root $x=x(y)$ in $(a,\infty)$ , and

2.2.2		$x(y)\sim y,$
$y\to\infty$ .
ⓘ Symbols: $\sim$ : asymptotic equality and $y$ : root Referenced by: §2.2 Permalink: http://dlmf.nist.gov/2.2.E2 Encodings: TeX, pMML, png See also: Annotations for §2.2 and Ch.2

Example

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Keywords:: Lagrange’s formula, Lagrange’s formula for reversion of series, asymptotic approximations and expansions, asymptotic approximations of integrals, asymptotic solutions of transcendental equations, reversion of, reversion of series
See also:: Annotations for §2.2 and Ch.2

2.2.3		$t^{2}-\ln t=y.$
ⓘ Symbols: $\ln\NVar{z}$ : principal branch of logarithm function and $y$ : root Permalink: http://dlmf.nist.gov/2.2.E3 Encodings: TeX, pMML, png See also: Annotations for §2.2, §2.2 and Ch.2

With $x=t^{2}$ , $f(x)=x-\frac{1}{2}\ln x$ . We may take $a=\frac{1}{2}$ . From (2.2.2)

2.2.4		$t=y^{\frac{1}{2}}\left(1+o\left(1\right)\right),$
$y\to\infty$ .
ⓘ Symbols: $o\left(\NVar{x}\right)$ : order less than and $y$ : root Permalink: http://dlmf.nist.gov/2.2.E4 Encodings: TeX, pMML, png See also: Annotations for §2.2, §2.2 and Ch.2

Higher approximations are obtainable by successive resubstitutions. For example

2.2.5		$t^{2}=y+\ln t=y+\tfrac{1}{2}\ln y+o\left(1\right),$
ⓘ Symbols: $o\left(\NVar{x}\right)$ : order less than, $\ln\NVar{z}$ : principal branch of logarithm function and $y$ : root Permalink: http://dlmf.nist.gov/2.2.E5 Encodings: TeX, pMML, png See also: Annotations for §2.2, §2.2 and Ch.2

and hence

2.2.6		$t=y^{\frac{1}{2}}\left(1+\tfrac{1}{4}y^{-1}\ln y+o\left(y^{-1}\right)\right),$
$y\to\infty$ .
ⓘ Symbols: $o\left(\NVar{x}\right)$ : order less than, $\ln\NVar{z}$ : principal branch of logarithm function and $y$ : root Permalink: http://dlmf.nist.gov/2.2.E6 Encodings: TeX, pMML, png See also: Annotations for §2.2, §2.2 and Ch.2

An important case is the reversion of asymptotic expansions for zeros of special functions. In place of (2.2.1) assume that

2.2.7		$f(x)\sim x+f_{0}+f_{1}x^{-1}+f_{2}x^{-2}+\cdots,$
$x\to\infty$ .
ⓘ Symbols: $\sim$ : Poincaré asymptotic expansion, $f(x)$ : function and $f_{s}$ : coefficients Permalink: http://dlmf.nist.gov/2.2.E7 Encodings: TeX, pMML, png See also: Annotations for §2.2, §2.2 and Ch.2

Then

2.2.8		$x\sim y-F_{0}-F_{1}y^{-1}-F_{2}y^{-2}-\cdots,$
$y\to\infty$ ,
ⓘ Symbols: $\sim$ : Poincaré asymptotic expansion, $y$ : root and $F_{s}$ : coefficients Permalink: http://dlmf.nist.gov/2.2.E8 Encodings: TeX, pMML, png See also: Annotations for §2.2, §2.2 and Ch.2

where $F_{0}=f_{0}$ and $sF_{s}$ ( $s\geq 1$ ) is the coefficient of $x^{-1}$ in the asymptotic expansion of $(f(x))^{s}$ (Lagrange’s formula for the reversion of series). Conditions for the validity of the reversion process in $\mathbb{C}$ are derived in Olver (1997b, pp. 14–16). Applications to real and complex zeros of Airy functions are given in Fabijonas and Olver (1999). For other examples see de Bruijn (1961, Chapter 2).

2.1 Definitions and Elementary Properties 2.3 Integrals of a Real Variable

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§2.2 Transcendental Equations

Example

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