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21: 9.7 Asymptotic Expansions

§9.7 Asymptotic Expansions

… ►Numerical values of $\chi (n)$ are given in Table 9.7.1 for $n=1(1)20$ to 2D. … ►

§9.7(iii) Error Bounds for Real Variables

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§9.7(iv) Error Bounds for Complex Variables

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22: 5.11 Asymptotic Expansions

§5.11 Asymptotic Expansions

… ►and … ►Wrench (1968) gives exact values of $g_k$ up to $g_{20}$. … ►If $z$ is complex, then the remainder terms are bounded in magnitude by ${\sec }^{2n}\left(\frac{1}{2}\mathrm{ph}z\right)$ for (5.11.1), and ${\sec }^{2n+1}\left(\frac{1}{2}\mathrm{ph}z\right)$ for (5.11.2), times the first neglected terms. … ►

23: Bibliography S

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Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.

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External Links:

ISSN 0036-1410, Document, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§22.19(i), §22.20(vi).

Encodings:

BibTeX

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Very high accuracy Chebyshev expansions for the basic trigonometric functions. Math. Comp. 34 (149), pp. 237–244.

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External Links:

ISSN 0025-5718, FullText, Document, MathReview (Claude Carasso), ZentralBlatt

Referenced by:

§4.47(i).

Encodings:

BibTeX

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Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

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External Links:

Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§28.8(iv).

Encodings:

BibTeX

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A Maple package for symmetric functions. J. Symbolic Comput. 20 (5-6), pp. 755–768.

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Notes:

For software see John Stembridge’s home page. The SF package contains Maple software for zonal, Jack, and Macdonald polynomials.

External Links:

Home Page, ISSN 0747-7171, Document, MathReview, ZentralBlatt

Referenced by:

3rd item.

Encodings:

BibTeX

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Numerical Methods Based on Sinc and Analytic Functions. Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.

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Notes:

Sinc-Pack, a set of 480 Matlab programs with a 470-page tutorial, is available for purchase from SINC, LLC by contacting Frank Stenger.

External Links:

ISBN 0-387-94008-1, MathReview (B. Boyanov), ZentralBlatt

Referenced by:

§3.3(vi), §3.4(i), §3.5(i).

Encodings:

BibTeX

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24: 12.10 Uniform Asymptotic Expansions for Large Parameter

… ►The turning points can be included if expansions in terms of Airy functions are used instead of elementary functions (§2.8(iii)). … ►Lastly, the function $g(\mu )$ in (12.10.3) and (12.10.4) has the asymptotic expansion: … ►

§12.10(vi) Modifications of Expansions in Elementary Functions

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Modified Expansions

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25: 36.2 Catastrophes and Canonical Integrals

… ► ${\mathrm{\Psi }}_1$ is related to the Airy function (§9.2): … … ►For the Bessel function $J$ see §10.2(ii). … ►Addendum: For further special cases see §36.2(iv) … ►

36.2.29 $${\mathrm{\Psi }}^{(\mathrm{H})}\left(0,0,z\right)=\overline{{\mathrm{\Psi }}^{(\mathrm{H})}\left(0,0,-z\right)}=\frac{2^{1/3}}{\sqrt{3}}\exp \left(\frac{1}{27}\mathrm{i}{z}^3\right){\mathrm{\Psi }}^{(\mathrm{E})}\left(0,0,-\frac{z}{2^{2/3}}\right),$$ .

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Symbols:

$\overline{z}$: complex conjugate, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\exp z$: exponential function, ${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $\mathrm{i}$: imaginary unit and $z$: real parameter

Referenced by:

Erratum (V1.0.5) for Chapters 8, 20, 36

Permalink:

http://dlmf.nist.gov/36.2.E29

Encodings:

pMML, png, TeX

Addition (effective with 1.0.5):

This equation has been added. For the proof see Berry and Howls (2010).

See also:

Annotations for §36.2(iv), §36.2 and Ch.36

26: 18.5 Explicit Representations

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§18.5(i) Trigonometric Functions

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Chebyshev

►With $x=\cos \theta =\frac{1}{2}(z+z^{-1})$, … ►

§18.5(iii) Finite Power Series, the Hypergeometric Function, and Generalized Hypergeometric Functions

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Hermite

…

27: 22.3 Graphics

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§22.3(i) Real Variables: Line Graphs

►Line graphs of the functions $\mathrm{sn}(x,k)$, $\mathrm{cn}(x,k)$, $\mathrm{dn}(x,k)$, $\mathrm{cd}(x,k)$, $\mathrm{sd}(x,k)$, $\mathrm{nd}(x,k)$, $\mathrm{dc}(x,k)$, $\mathrm{nc}(x,k)$, $\mathrm{sc}(x,k)$, $\mathrm{ns}(x,k)$, $\mathrm{ds}(x,k)$, and $\mathrm{cs}(x,k)$ for representative values of real $x$ and real $k$ illustrating the near trigonometric ($k=0$), and near hyperbolic ($k=1$) limits. … ►

§22.3(iii) Complex $z$; Real $k$

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§22.3(iv) Complex $k$

… ►In Figures 22.3.24 and 22.3.25, height corresponds to the absolute value of the function and color to the phase. …