reversion of series

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1: 2.2 Transcendental Equations

… ►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

… ►it being understood that these equalities are not reversible. … ►Let $\sum a_s{x}^{-s}$ be a formal power series (convergent or divergent) and for each positive integer $n$, … ►Most operations on asymptotic expansions can be carried out in exactly the same manner as for convergent power series. …For reversion see §2.2. … ►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. …

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 $𝒫$, then we may reverse the labeling of the $z_j$ along $𝒫$ 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

… ►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. … ►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 $\vartheta \left(x\right)$. …

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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External Links:

ISSN 1095-7200, Document, MathReview (Eberhard Wagner), ZentralBlatt

Referenced by:

§2.2, §2.2, §3.8(v), §9.9(iv), §9.9(iv).

Encodings:

BibTeX

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Basic Hypergeometric Series and Applications. Mathematical Surveys and Monographs, Vol. 27, American Mathematical Society, Providence, RI.

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Notes:

With a foreword by George E. Andrews

External Links:

ISBN 0-8218-1524-5, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§17.1, §17.16, §17.6(ii), Ch.17, Notations F.

Encodings:

BibTeX

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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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Notes:

Reprint with corrections of two books published originally in 1916 and 1936.

External Links:

MathReview

Referenced by:

§2.10(iii).

Encodings:

BibTeX

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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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External Links:

ISSN 0030-8730, MathReview (Paolo M. Soardi), ZentralBlatt

Referenced by:

§1.8(i).

Encodings:

BibTeX

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Series expansions of symmetric elliptic integrals. Math. Comp. 81 (278), pp. 957–990.

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External Links:

Document, ZentralBlatt

Referenced by:

§20.7(ii), §20.7(ii).

Encodings:

BibTeX

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7: 17.6 ${}_2{}^{}\varphi _1^{}$ Function

… ►Note that for several of the equations below, the constraints are included to guarantee that the infinite series representation (17.4.1) of the ${}_2{}^{}\varphi _1^{}$ functions converges. … ►This reverses the order of summation in (17.6.2): …