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11: 3.11 Approximation Techniques

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§3.11(iii) Minimax Rational Approximations

… ►Then the minimax (or best uniform) rational approximation … ► ►

Example

… ►The rational function …

12: 32.8 Rational Solutions

§32.8 Rational Solutions

… ►Special rational solutions of ${\text{P}}_{\text{III}}$ are … ►Then ${\text{P}}_{\text{III}}$ has rational solutions iff … ►These rational solutions have the form … ►

13: 3.1 Arithmetics and Error Measures

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§3.1(iii) Rational Arithmetics

►Computer algebra systems use exact rational arithmetic with rational numbers $p/q$, where $p$ and $q$ are multi-length integers. During the calculations common divisors are removed from the rational numbers, and the final results can be converted to decimal representations of arbitrary length. …

14: 11.15 Approximations

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§11.15(ii) Rational and Polynomial Approximations

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Newman (1984) gives polynomial approximations for ${𝐇}_n\left(x\right)$ for $n=0,1$, $0\le x\le 3$, and rational-fraction approximations for ${𝐇}_n\left(x\right)-Y_n\left(x\right)$ for $n=0,1$, $x\ge 3$. The maximum errors do not exceed 1.2×10⁻⁸ for the former and 2.5×10⁻⁸ for the latter.

15: 13.31 Approximations

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§13.31(iii) Rational Approximations

►In Luke (1977a) the following rational approximation is given, together with its rate of convergence. …

16: Mathematical Introduction

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$(a,b]$ or $[a,b)$	half-closed intervals.
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$\mathbb{Q}$	set of all rational numbers.
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17: 8.27 Approximations

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Luke (1975, p. 106) gives rational and Padé approximations, with remainders, for $E_1\left(z\right)$ and $z^{-1}{\int }_0^z{t}^{-1}(1-{\mathrm{e}}^{-t})dt$ for complex $z$ with $\left|\mathrm{ph}z\right|\le \pi$.

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Verbeeck (1970) gives polynomial and rational approximations for $E_p\left(x\right)=({\mathrm{e}}^{-x}/x)P(z)$, approximately, where $P(z)$ denotes a quotient of polynomials of equal degree in $z=x^{-1}$.

18: Bibliography Y

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On rational solutions of the second Painlevé equation. Vesti Akad. Navuk. BSSR Ser. Fiz. Tkh. Nauk. 3, pp. 30–35 (Russian).

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Referenced by:

§32.8(ii).

Encodings:

BibTeX

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19: 24.10 Arithmetic Properties

… ►Here and elsewhere two rational numbers are congruent if the modulus divides the numerator of their difference. …

20: Bibliography V

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Integrating products of Bessel functions with an additional exponential or rational factor. Comput. Phys. Comm. 178 (8), pp. 578–590.

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Notes:

Associated computer program available online

External Links:

ISSN 0010-4655, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§10.74(vii), §10.74(vii).

Encodings:

BibTeX

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Rational approximations for exponential integrals $E_n(x)$ . Acad. Roy. Belg. Bull. Cl. Sci. (5) 56, pp. 1064–1072.

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External Links:

MathReview (A. Magnus), ZentralBlatt

Referenced by:

3rd item.

Encodings:

BibTeX

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On the rational solutions of the second Painlevé equation. Differ. Uravn. 1 (1), pp. 79–81 (Russian).

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Notes:

In Russian. English translation: Differential Equations 1(1), pp. 58–59.

External Links:

MathReview, ZentralBlatt

Referenced by:

§32.8(ii).

Encodings:

BibTeX

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