# reversion of series

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## 7 matching pages

##### 1: 2.2 Transcendental Equations

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►where ${F}_{0}={f}_{0}$ and $s{F}_{s}$ ($s\ge 1$) is the coefficient of ${x}^{-1}$ in the asymptotic expansion of ${(f(x))}^{s}$ (

*Lagrange’s formula for the reversion of series*). …##### 2: 2.1 Definitions and Elementary Properties

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►it being understood that these equalities are not reversible.
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►Let $\sum {a}_{s}{x}^{-s}$ be a formal power series (convergent or divergent) and for each positive integer $n$,
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►Most operations on asymptotic expansions can be carried out in exactly the same manner as for convergent power series.
…For reversion see §2.2.
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►In those cases it is usually necessary to interpret each infinite series separately in the manner described above; that is, it is not always possible to reinterpret the asymptotic approximation as a single asymptotic expansion.
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##### 3: 3.7 Ordinary Differential Equations

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###### §3.7(ii) Taylor-Series Method: Initial-Value Problems

… ►Similarly, if $w(z)$ is decaying at least as fast as all other solutions along $\mathcal{P}$, then we may reverse the labeling of the ${z}_{j}$ along $\mathcal{P}$ and begin with initial values $w(b)$ and ${w}^{\prime}(b)$. ►###### §3.7(iii) Taylor-Series Method: Boundary-Value Problems

… ►It will be observed that the present formulation of the Taylor-series method permits considerable parallelism in the computation, both for initial-value and boundary-value problems. … ►General methods for boundary-value problems for ordinary differential equations are given in Ascher et al. (1995). …##### 4: 27.13 Functions

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►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.
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►Mordell (1917) notes that ${r}_{k}\left(n\right)$ is the coefficient of ${x}^{n}$ in the power-series expansion of the $k$th power of the series for $\vartheta \left(x\right)$.
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##### 5: 1.7 Inequalities

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###### Minkowski’s Inequality

… ►The direction of the inequality is reversed, that is, $\ge $, when $$. … ►###### Minkowski’s Inequality

… ►The direction of the inequality is reversed, that is, $\ge $, when $$. …##### 6: Bibliography F

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On the reversion of an asymptotic expansion and the zeros of the Airy functions.
SIAM Rev. 41 (4), pp. 762–773.
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Basic Hypergeometric Series and Applications.
Mathematical Surveys and Monographs, Vol. 27, American Mathematical Society, Providence, RI.
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Studies on Divergent Series and Summability & The Asymptotic Developments of Functions Defined by Maclaurin Series.
Chelsea Publishing Co., New York.
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Asymptotic expansions of the Lebesgue constants for Jacobi series.
Pacific J. Math. 122 (2), pp. 391–415.
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Series expansions of symmetric elliptic integrals.
Math. Comp. 81 (278), pp. 957–990.
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