approximations (except asymptotic)

SearchAdvancedHelp

(0.002 seconds)

11—19 of 19 matching pages

11: Bibliography K

… ►

Asymptotic approximations for the first incomplete elliptic integral near logarithmic singularity. J. Comput. Appl. Math. 205 (1), pp. 186–206.

ⓘ

External Links:

ISSN 0377-0427, Document, FullText, MathReview (José Luis López), ZentralBlatt

Referenced by:

§19.12, §19.12, §19.36(iv), §19.36(iv).

Encodings:

BibTeX

… ►

Uniform asymptotic approximations for the Meixner-Sobolev polynomials. Anal. Appl. (Singap.) 10 (3), pp. 345–361.

ⓘ

External Links:

ISSN 0219-5305, Document, FullText, MathReview (Diego E. Dominici), ZentralBlatt

Referenced by:

§18.24, §18.24.

Encodings:

BibTeX

►

Exponentially accurate uniform asymptotic approximations for integrals and Bleistein’s method revisited. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 469 (2153), pp. 20130008, 12.

ⓘ

External Links:

ISSN 1364-5021, Document, FullText, MathReview (Ulrich D. Jentschura), ZentralBlatt

Referenced by:

§2.4(vi), §2.4(vi).

Encodings:

BibTeX

… ►

Approximation Formulae for Generalized Hypergeometric Functions for Large Values of the Parameters. J. B. Wolters, Groningen.

ⓘ

External Links:

MathReview (L. J. Slater), ZentralBlatt

Referenced by:

§16.11(iii).

Encodings:

BibTeX

… ►

Zeros of exceptional Hermite polynomials. J. Approx. Theory 200, pp. 28–39.

ⓘ

External Links:

ISSN 0021-9045, Document, Link, MathReview (Mario Pérez Riera)

Referenced by:

§18.39(i).

Encodings:

BibTeX

…

12: 3.6 Linear Difference Equations

… ►Then $w_n$ is said to be a recessive (equivalently, minimal or distinguished) solution as $n\to \mathrm{\infty }$, and it is unique except for a constant factor. … ►The values of $w_N$ and $w_{N+1}$ needed to begin the backward recursion may be available, for example, from asymptotic expansions (§2.9). … ►(It should be observed that for $n>10$, however, the $w_n$ are progressively poorer approximations to ${𝐄}_n\left(1\right)$: the underlined digits are in error.) … ►or for systems of $k$ first-order inhomogeneous equations, boundary-value methods are the rule rather than the exception. …Here $\mathrm{\ell }\in [0,k]$, and its actual value depends on the asymptotic behavior of the wanted solution in relation to those of the other solutions. …

13: 6.18 Methods of Computation

… ►However, this problem is less severe for the series of spherical Bessel functions because of their more rapid rate of convergence, and also (except in the case of (6.10.6)) absence of cancellation when $z=x$ ($>0$). ►For large $x$ and $\left|z\right|$, expansions in inverse factorial series (§6.10(i)) or asymptotic expansions (§6.12) are available. The attainable accuracy of the asymptotic expansions can be increased considerably by exponential improvement. … ►Power series, asymptotic expansions, and quadrature can also be used to compute the functions $\mathrm{f}\left(z\right)$ and $\mathrm{g}\left(z\right)$. … ►Zeros of $\mathrm{Ci}\left(x\right)$ and $\mathrm{si}\left(x\right)$ can be computed to high precision by Newton’s rule (§3.8(ii)), using values supplied by the asymptotic expansion (6.13.2) as initial approximations. …

14: Bibliography N

… ►

On an asymptotic expansion of the Kontorovich-Lebedev transform. Applicable Anal. 39 (4), pp. 249–263.

ⓘ

External Links:

ISSN 0003-6811, Document, MathReview (Aloknath Chakrabarti), ZentralBlatt

Referenced by:

§10.43(v).

Encodings:

BibTeX

… ►

Mathematical Approximation of Special Functions. Nova Science Publishers Inc., Commack, NY.

ⓘ

Notes:

Ten papers on Chebyshev expansions

External Links:

ISBN 1-56072-052-2, MathReview (N. Hayek Calil), ZentralBlatt

Referenced by:

§10.76(ii), §10.76(ii), §10.76(ii), §10.76(iii), 2nd item.

Encodings:

BibTeX

►

Uniform Asymptotic Approximations of Solutions of Second-order Linear Differential Equations, with a Coalescing Simple Turning Point and Simple Pole. Ph.D. Thesis, University of Maryland, College Park, MD.

ⓘ

Referenced by:

§2.8(vi).

Encodings:

BibTeX

… ►

Approximations for the Bessel and Struve functions. Math. Comp. 43 (168), pp. 551–556.

ⓘ

External Links:

ISSN 0025-5718, FullText, MathReview (S. Conde), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

… ►

Die Gammafunktion. Band I. Handbuch der Theorie der Gammafunktion. Band II. Theorie des Integrallogarithmus und verwandter Transzendenten. Chelsea Publishing Co., New York (German).

ⓘ

Notes:

Both books are textually unaltered except for correction of errata.

External Links:

MathReview

Referenced by:

N. Nielsen (1906a), N. Nielsen (1906b).

Encodings:

BibTeX

…

15: 9.12 Scorer Functions

… ►

§9.12(viii) Asymptotic Expansions

… ►For other phase ranges combine these results with the connection formulas (9.12.11)–(9.12.14) and the asymptotic expansions given in §9.7. … ►

Integrals

… ►

9.12.31 $${\int }_0^z\mathrm{Hi}\left(-t\right)dt\sim \frac{1}{\pi }\ln z+\frac{2\gamma +\ln 3}{3\pi }+\frac{1}{\pi }\sum _{k=1}^{\mathrm{\infty }}{(-1)}^{k-1}\frac{(3k-1)!}{k!{(3z^3)}^k},$$ $|\mathrm{ph}z|\le \frac{2}{3}\pi -\delta$,

ⓘ

Symbols:

$\gamma$: Euler’s constant, $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $k$: nonnegative integer, $x$: real variable, $z$: complex variable and $\delta$: small positive constant

Proof sketch:

Except for the constant term, this be verified by termwise integration of (9.12.27). To evaluate the constant term replace $z$ by $-x$ ($\le 0$) in (9.12.20) and integrate (§1.5(v)) to obtain $\pi {\int }_0^x\mathrm{Hi}\left(-t\right)dt={\int }_0^{\mathrm{\infty }}(1-{\mathrm{e}}^{-xt}){\mathrm{e}}^{-\frac{1}{3}{t}^3}{t}^{-1}dt$. Next, integrate the right-hand side of this equation by parts—integrating the factor $t^{-1}$ and differentiating the rest. As $x\to \mathrm{\infty }$ the asymptotic expansions of ${\int }_0^{\mathrm{\infty }}x{\mathrm{e}}^{-xt}{\mathrm{e}}^{-\frac{1}{3}{t}^3}(\ln t)dt$ and ${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-xt}{t}^2{\mathrm{e}}^{-\frac{1}{3}{t}^3}(\ln t)dt$ follow from (2.3.9). Also, ${\int }_0^{\mathrm{\infty }}{t}^2{\mathrm{e}}^{-\frac{1}{3}{t}^3}(\ln t)dt$ can be found by replacing $\frac{1}{3}{t}^3$ by $t$ and referring to the first of (5.9.18). (This equation first appeared in Rothman (1954a). As noted in this reference these results were derived by the author of the present DLMF chapter, but the proof was not included.)

Referenced by:

(9.12.30)

Permalink:

http://dlmf.nist.gov/9.12.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(viii), §9.12(viii), §9.12 and Ch.9

… ►For the above properties and further results, including the distribution of complex zeros, asymptotic approximations for the numerically large real or complex zeros, and numerical tables see Gil et al. (2003c). …

16: 18.16 Zeros

… ►except when ${\alpha }^2={\beta }^2=\frac{1}{4}$. … ►

Asymptotic Behavior

… ►

Asymptotic Behavior

… ►For an error bound for the first approximation yielded by this expansion see Olver (1997b, p. 408). ►Lastly, in view of (18.7.19) and (18.7.20), results for the zeros of $L_n^{(\pm \frac{1}{2})}\left(x\right)$ lead immediately to results for the zeros of $H_n\left(x\right)$. …

17: 3.8 Nonlinear Equations

… ►For real functions $f(x)$ the sequence of approximations to a real zero $\xi$ will always converge (and converge quadratically) if either: … ►Inverse linear interpolation (§3.3(v)) is used to obtain the first approximation: … ►The method converges locally and quadratically, except when the wanted quadratic factor is a multiple factor of $q(z)$. … ►Initial approximations to the zeros can often be found from asymptotic or other approximations to $f(z)$, or by application of the phase principle or Rouché’s theorem; see §1.10(iv). … ►For describing the distribution of complex zeros of solutions of linear homogeneous second-order differential equations by methods based on the Liouville–Green (WKB) approximation, see Segura (2013). …

18: 25.10 Zeros

… ►Except for the trivial zeros, $\zeta \left(s\right)\ne 0$ for $\mathrm{\Re }s\le 0$. … ►The error term $R(t)$ can be expressed as an asymptotic series that begins ►

25.10.4 $$R(t)={(-1)}^{m-1}{\left(\frac{2\pi }{t}\right)}^{1/4}\frac{\cos \left(t-(2m+1)\sqrt{2\pi t}-\frac{1}{8}\pi \right)}{\cos \left(\sqrt{2\pi t}\right)}+O\left(t^{-3/4}\right).$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function and $m$: nonnegative integer

Keywords:

asymptoticapproximation

Source:

Titchmarsh (1986b, (4.17.5), p. 89)

Referenced by:

§25.10(ii), Erratum (V1.1.4) for Subsection 25.10(ii)

Permalink:

http://dlmf.nist.gov/25.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §25.10(ii), §25.10 and Ch.25

…

19: Bibliography L

… ►

Asymptotic Analysis. Mathematical Centre Tracts, Mathematisch Centrum, Amsterdam.

ⓘ

External Links:

MathReview (C. Comstock), ZentralBlatt

Referenced by:

Ch.2.

Encodings:

BibTeX

… ►

The spectral analysis of three families of exceptional Laguerre polynomials. J. Approx. Theory 202, pp. 5–41.

ⓘ

External Links:

ISSN 0021-9045, Document, Link, MathReview (Edmundo J. Huertas)

Referenced by:

§18.36(vi).

Encodings:

BibTeX

… ►

Approximations for elliptic integrals. Math. Comp. 22 (103), pp. 627–634.

ⓘ

External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§19.38.

Encodings:

BibTeX

… ►

Further approximations for elliptic integrals. Math. Comp. 24 (109), pp. 191–198.

ⓘ

External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§19.38.

Encodings:

BibTeX

… ►

Algorithms for rational approximations for a confluent hypergeometric function. Utilitas Math. 11, pp. 123–151.

ⓘ

External Links:

MathReview (F. Gotze), ZentralBlatt

Referenced by:

§13.31(iii).

Encodings:

BibTeX

…