Let be continuous and strictly increasing when and
Then for the equation has a unique root in , and
With , . We may take . From (2.2.2)
Higher approximations are obtainable by successive resubstitutions. For example
An important case is the reversion of asymptotic expansions for zeros of special functions. In place of (2.2.1) assume that
where and () is the coefficient of in the asymptotic expansion of (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).