# reversion of

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##### 1: 2.2 Transcendental Equations
2.2.6 $t=y^{\frac{1}{2}}\left(1+\tfrac{1}{4}y^{-1}\ln y+o\left(y^{-1}\right)\right),$ $y\to\infty$.
An important case is the reversion of asymptotic expansions for zeros of special functions. …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). …
##### 2: 13.22 Zeros
Asymptotic approximations to the zeros when the parameters $\kappa$ and/or $\mu$ are large can be found by reversion of the uniform approximations provided in §§13.20 and 13.21. …
##### 3: 1.7 Inequalities
The direction of the inequality is reversed, that is, $\geq$, when $0. … The direction of the inequality is reversed, that is, $\geq$, when $0. …
##### 4: 1.6 Vectors and Vector-Valued Functions
If $h(a)=b^{\prime}$ and $h(b)=a^{\prime}$, then the reparametrization is orientation-reversing and … A parametrization $\boldsymbol{{\Phi}}(u,v)$ of an oriented surface $S$ is orientation preserving if $\mathbf{T}_{u}\times\mathbf{T}_{v}$ has the same direction as the chosen normal at each point of $S$, otherwise it is orientation reversing. If $\boldsymbol{{\Phi}}_{1}$ and $\boldsymbol{{\Phi}}_{2}$ are both orientation preserving or both orientation reversing parametrizations of $S$ defined on open sets $D_{1}$ and $D_{2}$ respectively, then …
##### 5: 8.10 Inequalities
The inequalities in (8.10.1) and (8.10.2) are reversed when $a\geq 1$. …
##### 6: 12.11 Zeros
For large negative values of $a$ the real zeros of $U\left(a,x\right)$, $U'\left(a,x\right)$, $V\left(a,x\right)$, and $V'\left(a,x\right)$ can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii). …
##### 7: 27.13 Functions
In fact, there are four representations, given by $5=2^{2}+1^{2}=2^{2}+(-1)^{2}=(-2)^{2}+1^{2}=(-2)^{2}+(-1)^{2}$, and four more with the order of summands reversed. …
##### 8: 2.1 Definitions and Elementary Properties
it being understood that these equalities are not reversible. … For reversion see §2.2. …
##### 9: Bibliography F
• B. R. Fabijonas and F. W. J. Olver (1999) On the reversion of an asymptotic expansion and the zeros of the Airy functions. SIAM Rev. 41 (4), pp. 762–773.
• ##### 10: Bibliography D
• T. M. Dunster (2001c) Uniform asymptotic expansions for the reverse generalized Bessel polynomials, and related functions. SIAM J. Math. Anal. 32 (5), pp. 987–1013.