§2.2 Transcendental Equations

Notes:: See Olver (1997b, pp. 11–16) and Fabijonas and Olver (1999).
Keywords:: asymptotic solutions of transcendental equations, transcendental equations
Referenced by:: §10.70, §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 $f(x)\sim x,$ $x\to\infty$ .

Symbols:: $\sim$ : asymptotic equality and $f(x)$ : function
Referenced by:: ¶ ‣ §2.2
Permalink:: http://dlmf.nist.gov/2.2.E1
Encodings:: TeX, pMML, png

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

¶ Example

Keywords:: Lagrange’s formula for reversion of series, asymptotic approximations and expansions, asymptotic approximations of integrals, asymptotic solutions of transcendental equations, reversion of series

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

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $y$ : root
Permalink:: http://dlmf.nist.gov/2.2.E3
Encodings:: TeX, pMML, png

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

2.2.4 $t=y^{{\frac{1}{2}}}\left(1+\mathop{o\/}\nolimits\!\left(1\right)\right),$ $y\to\infty$ .

Symbols:: $\mathop{o\/}\nolimits\!\left(x\right)$ : order less than and $y$ : root
Permalink:: http://dlmf.nist.gov/2.2.E4
Encodings:: TeX, pMML, png

Higher approximations are obtainable by successive resubstitutions. For example

2.2.5 $t^{2}=y+\mathop{\ln\/}\nolimits t=y+\tfrac{1}{2}\mathop{\ln\/}\nolimits y+\mathop{o\/}\nolimits\!\left(1\right),$

Symbols:: $\mathop{o\/}\nolimits\!\left(x\right)$ : order less than, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $y$ : root
Permalink:: http://dlmf.nist.gov/2.2.E5
Encodings:: TeX, pMML, png

and hence

2.2.6 $t=y^{{\frac{1}{2}}}\left(1+\tfrac{1}{4}y^{{-1}}\mathop{\ln\/}\nolimits y+\mathop{o\/}\nolimits\!\left(y^{{-1}}\right)\right),$ $y\to\infty$ .

Symbols:: $\mathop{o\/}\nolimits\!\left(x\right)$ : order less than, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $y$ : root
Permalink:: http://dlmf.nist.gov/2.2.E6
Encodings:: TeX, pMML, 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 $f(x)\sim x+f_{0}+f_{1}x^{{-1}}+f_{2}x^{{-2}}+\cdots,$ $x\to\infty$ .

Symbols:: $\sim$ : asymptotic equality, $f(x)$ : function and $f_{s}$ : coefficients
Permalink:: http://dlmf.nist.gov/2.2.E7
Encodings:: TeX, pMML, png

Then

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

Symbols:: $\sim$ : asymptotic equality, $y$ : root and $F_{s}$ : coefficients
Permalink:: http://dlmf.nist.gov/2.2.E8
Encodings:: TeX, pMML, png

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).