# reversion of series

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## 6 matching pages

##### 1: 2.2 Transcendental Equations
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). …
##### 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
###### §3.7(ii) Taylor-Series Method: Initial-Value Problems
Similarly, if $w(z)$ is decaying at least as fast as all other solutions along $\mathscr{P}$, then we may reverse the labeling of the $z_{j}$ along $\mathscr{P}$ 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
###### Minkowski’s Inequality
The direction of the inequality is reversed, that is, $\geq$, when $0. …
###### Minkowski’s Inequality
The direction of the inequality is reversed, that is, $\geq$, when $0. …
##### 6: 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.
• N. J. Fine (1988) Basic Hypergeometric Series and Applications. Mathematical Surveys and Monographs, Vol. 27, American Mathematical Society, Providence, RI.
• W. B. Ford (1960) Studies on Divergent Series and Summability & The Asymptotic Developments of Functions Defined by Maclaurin Series. Chelsea Publishing Co., New York.
• C. L. Frenzen and R. Wong (1986) Asymptotic expansions of the Lebesgue constants for Jacobi series. Pacific J. Math. 122 (2), pp. 391–415.
• T. Fukushima (2012) Series expansions of symmetric elliptic integrals. Math. Comp. 81 (278), pp. 957–990.