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§2.2 Transcendental Equations

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

2.2.1 f(x)x,
x.

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

2.2.2 x(y)y,
y.

Example

2.2.3 t2-lnt=y.

With x=t2, f(x)=x-12lnx. We may take a=12. From (2.2.2)

2.2.4 t=y12(1+o(1)),
y.

Higher approximations are obtainable by successive resubstitutions. For example

2.2.5 t2=y+lnt=y+12lny+o(1),

and hence

2.2.6 t=y12(1+14y-1lny+o(y-1)),
y.

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)x+f0+f1x-1+f2x-2+,
x.

Then

2.2.8 xy-F0-F1y-1-F2y-2-,
y,

where F0=f0 and sFs (s1) 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 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).