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

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Symbols:: $o\left(\NVar{x}\right)$ : order less than, $\ln\NVar{z}$ : principal branch of logarithm function and $y$ : root
Permalink:: http://dlmf.nist.gov/2.2.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §2.2, §2.2 and Ch.2

►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 <mrow> <mn>0</mn> <mo><</mo> <mi href=

. … ►The direction of the inequality is reversed, that is,

\geq

, when

0 <mrow> <mn>0</mn> <mo><</mo> <mi href=

. …

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: 18.34 Bessel Polynomials

… ► Sometimes the polynomials

\theta_{n}(x;a,b)

are called reverse Bessel polynomials. …

10: 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.

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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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