§ 2.2. Transcendental Equations

Notes:: See Olver (1997b, pp. 11–16) and Fabijonas and Olver (1999).
Referenced by:: §2.1(iii)
Permalink:: http://dlmf.nist.gov/2.2

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

2.2.1 $f(x)\sim x,$ $x\to\infty$ .

Defines:: $f(x)$ : function
Symbols:: $\sim$ : asymptotically equal
Referenced by:: §2.2
Permalink:: http://dlmf.nist.gov/2.2.E1
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Then for $y>f(a)$ the equation $f(x)=y$ has a unique root $x=x(y)$ in $(a,\infty)$ , and

2.2.2 $x(y)\sim y,$ $y\to\infty$ .

Defines:: $y$ : root
Symbols:: $\sim$ : asymptotically equal
Referenced by:: §2.2
Permalink:: http://dlmf.nist.gov/2.2.E2
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¶ Example

2.2.3 $t^{2}-\ln t=y.$

Symbols:: $y$ : root
Permalink:: http://dlmf.nist.gov/2.2.E3
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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:: $o\!\left(x\right)$ : order symbol and $y$ : root
Permalink:: http://dlmf.nist.gov/2.2.E4
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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:: $o\!\left(x\right)$ : order symbol and $y$ : root
Permalink:: http://dlmf.nist.gov/2.2.E5
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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:: $o\!\left(x\right)$ : order symbol and $y$ : root
Permalink:: http://dlmf.nist.gov/2.2.E6
Encodings:: TeX, pMathML, png

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 $f(x)\sim x+f_{0}+f_{1}x^{{-1}}+f_{2}x^{{-2}}+\cdots,$ $x\to\infty$ .

Defines:: $f_{s}$ : coefficients
Symbols:: $\sim$ : asymptotically equal and $f(x)$ : function
Permalink:: http://dlmf.nist.gov/2.2.E7
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Then

2.2.8 $x\sim y-F_{0}-F_{1}y^{{-1}}-F_{2}y^{{-2}}-\cdots,$ $y\to\infty$ ,

Defines:: $F_{s}$ : coefficients
Symbols:: $\sim$ : asymptotically equal and $y$ : root
Permalink:: http://dlmf.nist.gov/2.2.E8
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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 $\Complex$ 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).