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21: Bibliography W

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Rational approximations for the modified Bessel function of the second kind. Comput. Phys. Comm. 59 (3), pp. 471–493.

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External Links:

ISSN 0010-4655, Document, MathReview, ZentralBlatt

Referenced by:

§10.76(ii).

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A rational approximant for the digamma function. Numer. Algorithms 33 (1-4), pp. 499–507.

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Notes:

International Conference on Numerical Algorithms, Vol. I (Marrakesh, 2001)

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ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§3.9(vi), §5.23(i).

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Rational Chebyshev approximation. Numer. Math. 10 (4), pp. 289–306.

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External Links:

ISSN 0029-599X, Document, MathReview Entry, ZentralBlatt

Referenced by:

§3.11(iii).

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Rational function certification of multisum/integral/“$q$” identities. Bull. Amer. Math. Soc. (N.S.) 27 (1), pp. 148–153.

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ISSN 0273-0979, Pdf, Document, MathReview (Robert A. Gustafson), ZentralBlatt

Referenced by:

§16.4(iii).

Encodings:

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Rational Chebyshev approximations for the Bessel functions $J_0(x)$, $J_1(x)$, $Y_0(x)$, $Y_1(x)$ . Math. Comp. 39 (160), pp. 617–623.

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Notes:

Tables on microfiche

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ISSN 0025-5718, FullText, MathReview (C. W. Clenshaw), ZentralBlatt

Referenced by:

9th item.

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22: Bibliography J

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Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II. Phys. D 2 (3), pp. 407–448.

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ISSN 0167-2789, Document, MathReview (V. A. Golubeva), ZentralBlatt

Referenced by:

§32.4(i), §32.4(ii), §32.4(iv), §32.4(v), §32.6(ii), §32.6(iii), §32.6(iv), §32.6(v), §32.6(v), §32.6(v).

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REMES2 — a Fortran program to calculate rational minimax approximations to a given function. Technical Report Technical Report AECL-4210, Atomic Energy of Canada Limited. Chalk River Nuclear Laboratories, Chalk River, Ontario.

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Referenced by:

§3.11(iii).

Encodings:

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23: 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.

External Links:

ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§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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External Links:

MathReview Entry

Referenced by:

§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).

Encodings:

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Foundations of Finite Precision Rational Arithmetic. In Fundamentals of Numerical Computation (Computer-oriented Numerical Analysis), G. Alefeld and R. D. Grigorieff (Eds.), Comput. Suppl., Vol. 2, Vienna, pp. 85–111.

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External Links:

MathReview (G. Blanch), ZentralBlatt

Referenced by:

§3.1(iii).

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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

Referenced by:

§32.8(vi).

Encodings:

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24: 19.2 Definitions

… ►Let $s^2(t)$ be a cubic or quartic polynomial in $t$ with simple zeros, and let $r(s,t)$ be a rational function of $s$ and $t$ containing at least one odd power of $s$. … ►

19.2.1 $$\int r(s,t)dt$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral and $r(s,t)$: rational function

Referenced by:

§19.2(i)

Permalink:

http://dlmf.nist.gov/19.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §19.2(i), §19.2 and Ch.19

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19.2.2 $$r(s,t)=\frac{(p_1+p_{2}s)(p_3-p_{4}s)s}{(p_3+p_{4}s)(p_3-p_{4}s)s}=\frac{\rho }{s}+\sigma ,$$

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Symbols:

$r(s,t)$: rational function, $\rho (t)$: rational function, $\sigma (t)$: rational function and $p_j$: polynomial

Permalink:

http://dlmf.nist.gov/19.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.2(i), §19.2 and Ch.19

►where $p_j$ is a polynomial in $t$ while $\rho$ and $\sigma$ are rational functions of $t$. … ►

19.2.3 $$\int \frac{\rho (t)}{s(t)}dt.$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral and $\rho (t)$: rational function

Referenced by:

§19.14(ii), §19.16(ii), §19.29(ii)

Permalink:

http://dlmf.nist.gov/19.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §19.2(i), §19.2 and Ch.19

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25: 32.9 Other Elementary Solutions

… ►These are rational solutions in $\zeta =z^{1/3}$ of the form … ►These are rational solutions in $\zeta =z^{1/2}$ of the form …

26: Bibliography

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Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem. Comm. Pure Appl. Math. 30 (1), pp. 95–148.

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External Links:

MathReview (Alexander A. Pankov), ZentralBlatt

Referenced by:

§23.21(ii).

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Rational solutions of Painlevé equations. Stud. Appl. Math. 61 (1), pp. 31–53.

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External Links:

ISSN 0022-2526, MathReview (H. Hochstadt), ZentralBlatt

Referenced by:

§32.10(ii), §32.8(i), §32.8(v).

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A $q$-beta integral on the unit circle and some biorthogonal rational functions. Proc. Amer. Math. Soc. 121 (2), pp. 553–561.

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External Links:

ISSN 0002-9939, FullText, MathReview (S. P. Goyal), ZentralBlatt

Referenced by:

§18.33(v).

Encodings:

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Rational function approximations for Fermi-Dirac integrals. The Astrophysical Journal Supplement Series 84, pp. 101–108.

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Notes:

Double-precision Fortran, maximum precision 8D.

External Links:

ISSN 0067-0049, Document, Code

Referenced by:

5th item, 1st item.

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27: Bibliography I

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$q$-Hermite polynomials, biorthogonal rational functions, and $q$-beta integrals. Trans. Amer. Math. Soc. 346 (1), pp. 63–116.

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External Links:

ISSN 0002-9947, FullText, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.28(vii).

Encodings:

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28: 9.19 Approximations

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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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29: 31.14 General Fuchsian Equation

… ►An algorithm given in Kovacic (1986) determines if a given (not necessarily Fuchsian) second-order homogeneous linear differential equation with rational coefficients has solutions expressible in finite terms (Liouvillean solutions). …

30: 23.20 Mathematical Applications

… ►If $a,b\in \mathbb{Q}$, then by rescaling we may assume $a,b\in \mathbb{Z}$. Let $T$ denote the set of points on $C$ that are of finite order (that is, those points $P$ for which there exists a positive integer $n$ with $nP=o$), and let $I,K$ be the sets of points with integer and rational coordinates, respectively. …