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31: Bibliography C

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Uniform asymptotic expansion of an integral with a saddle point, a pole and a branch point. Proc. Roy. Soc. London Ser. A 426, pp. 273–286.

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ISSN 0080-4630, FullText, MathReview, ZentralBlatt

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§2.4(vi).

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Special polynomials associated with rational solutions of the fifth Painlevé equation. J. Comput. Appl. Math. 178 (1-2), pp. 111–129.

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ISSN 0377-0427, Document, MathReview (Davide Batic), ZentralBlatt

Referenced by:

§32.8(v), §32.9(ii).

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Rational Chebyshev approximations for the exponential integral $E_1(x)$ . Math. Comp. 22 (103), pp. 641–649.

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FullText, MathReview, ZentralBlatt

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2nd item.

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Rational Chebyshev approximations for the error function. Math. Comp. 23 (107), pp. 631–637.

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FullText, MathReview, ZentralBlatt

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2nd item.

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A survey of practical rational and polynomial approximation of functions. SIAM Rev. 12 (3), pp. 400–423.

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Document, MathReview (J. H. Curtiss), ZentralBlatt

Referenced by:

§3.11(i).

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32: 18.16 Zeros

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Asymptotic Behavior

… ►The constant $j_{\alpha ,m}^2$ in (18.16.10) is the best possible since the ratio of the two sides of this inequality tends to 1 as $n\to \mathrm{\infty }$. … ►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 $H_n\left(x\right)$. …

33: 9.19 Approximations

§9.19 Approximations

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§9.19(i) Approximations in Terms of Elementary Functions

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Moshier (1989, §6.14) provides minimax rational approximations for calculating $\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$, $\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$. They are in terms of the variable $\zeta$, where $\zeta =\frac{2}{3}{x}^{3/2}$ when $x$ is positive, $\zeta =\frac{2}{3}{(-x)}^{3/2}$ when $x$ is negative, and $\zeta =0$ when $x=0$. The approximations apply when , that is, when or . The precision in the coefficients is 21S.

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§9.19(ii) Expansions in Chebyshev Series

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§9.19(iii) Approximations in the Complex Plane

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34: Bibliography K

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Determinant structure of the rational solutions for the Painlevé II equation. J. Math. Phys. 37 (9), pp. 4693–4704.

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ISSN 0022-2488, Document, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.8(ii).

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Determinant structure of the rational solutions for the Painlevé IV equation. J. Phys. A 31 (10), pp. 2431–2446.

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ISSN 0305-4470, Document, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.8(iv).

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

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

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§18.24, §18.24.

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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.

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

Referenced by:

§2.4(vi), §2.4(vi).

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Rational solutions of the fifth Painlevé equation. Differential Integral Equations 7 (3-4), pp. 967–1000.

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ISSN 0893-4983, MathReview, ZentralBlatt

Referenced by:

§32.8(v).

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35: 36.15 Methods of Computation

… ►Close to the bifurcation set but far from $𝐱=\mathrm{𝟎}$, the uniform asymptotic approximations of §36.12 can be used. … ►This can be carried out by direct numerical evaluation of canonical integrals along a finite segment of the real axis including all real critical points of $\mathrm{\Phi }$, with contributions from the contour outside this range approximated by the first terms of an asymptotic series associated with the endpoints. …

36: Annie A. M. Cuyt

… ►Her main research interest is in the area of numerical approximation theory and its applications to a diversity of problems in scientific computing. …A lot of her research has been devoted to rational approximations, in one as well as in many variables, and sparse interpolation. …

37: 2.11 Remainder Terms; Stokes Phenomenon

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§2.11(iii) Exponentially-Improved Expansions

… ►One is uniformly valid for $-\pi +\delta \le \mathrm{ph}z\le 3\pi -\delta$ with bounded $|\alpha |$, and achieves uniform exponential improvement throughout $0\le \mathrm{ph}z\le \pi$: … ►In addition to achieving uniform exponential improvement, particularly in $|\mathrm{ph}z|\le \pi$ for $w_1(z)$, and $0\le \mathrm{ph}z\le 2\pi$ for $w_2(z)$, the re-expansions (2.11.20), (2.11.21) are resurgent. … ►

§2.11(vi) Direct Numerical Transformations

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38: 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. …

39: Bibliography M

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Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.

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Notes:

Includes Fortran code, maximum accuracy 20S.

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ISSN 1017-1398, Document, MathReview, ZentralBlatt

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§6.13, 4th item, §6.21(ii).

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New ladder operators for a rational extension of the harmonic oscillator and superintegrability of some two-dimensional systems. J. Math. Phys. 54 (10), pp. Paper 102102, 12 pp..

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MathReview Entry

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§18.36(vi).

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A determinant formula for a class of rational solutions of Painlevé V equation. Nagoya Math. J. 168, pp. 1–25.

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ISSN 0027-7630, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.8(v).

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Uniform computation of the error function and other related functions. Math. Comp. 25 (114), pp. 339–344.

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FullText, MathReview, ZentralBlatt

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§7.22(i).

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Rational solutions of the Painlevé VI equation. J. Phys. A 34 (11), pp. 2281–2294.

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ISSN 0305-4470, Document, MathReview (Shun Shimomura), ZentralBlatt

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§32.8(vi).

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40: Bibliography D

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Uniform asymptotic expansions for prolate spheroidal functions with large parameters. SIAM J. Math. Anal. 17 (6), pp. 1495–1524.

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ISSN 0036-1410, Document, MathReview (R. K. Gupta), ZentralBlatt

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§30.9(i).

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Uniform asymptotic approximation of Mathieu functions. Methods Appl. Anal. 1 (2), pp. 143–168.

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ISSN 1073-2772, Pdf, MathReview (J. M. H. Peters), ZentralBlatt

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§28.8(iv).

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Uniform asymptotic expansions for Charlier polynomials. J. Approx. Theory 112 (1), pp. 93–133.

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ISSN 0021-9045, Document, MathReview (Eberhard Wagner), ZentralBlatt

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§18.24.

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Uniform asymptotic approximations for the Whittaker functions $M_{\kappa ,i\mu }(z)$ and $W_{\kappa ,i\mu }(z)$ . Anal. Appl. (Singap.) 1 (2), pp. 199–212.

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ISSN 0219-5305, Document, MathReview (Hasan Taşeli), ZentralBlatt

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§13.20(v).

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Uniform asymptotic approximations for incomplete Riemann zeta functions. J. Comput. Appl. Math. 190 (1-2), pp. 339–353.

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ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§8.22(ii).

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