rational functions

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1: 16.26 Approximations

… ►For discussions of the approximation of generalized hypergeometric functions and the Meijer $G$-function in terms of polynomials, rational functions, and Chebyshev polynomials see Luke (1975, §§5.12 - 5.13) and Luke (1977b, Chapters 1 and 9).

2: 5.19 Mathematical Applications

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§5.19(i) Summation of Rational Functions

►As shown in Temme (1996b, §3.4), the results given in §5.7(ii) can be used to sum infinite series of rational functions. …

3: 4.47 Approximations

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§4.47(ii) Rational Functions

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4: 5.23 Approximations

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§5.23(i) Rational Approximations

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§5.23(iii) Approximations in the Complex Plane

►See Schmelzer and Trefethen (2007) for a survey of rational approximations to various scaled versions of $\mathrm{\Gamma }\left(z\right)$. …

5: 16.7 Relations to Other Functions

… ►Further representations of special functions in terms of ${}_p{}^{}F_q^{}$ functions are given in Luke (1969a, §§6.2–6.3), and an extensive list of ${}_{q+1}{}^{}F_q^{}$ functions with rational numbers as parameters is given in Krupnikov and Kölbig (1997).

6: 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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7: 3.11 Approximation Techniques

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§3.11(iii) Minimax Rational Approximations

… ►The theory of polynomial minimax approximation given in §3.11(i) can be extended to the case when $p_n(x)$ is replaced by a rational function $R_{k,\mathrm{\ell }}(x)$. … ►

Example

… ►The rational function … ►A general procedure is to approximate $F$ by a rational function $R$ (vanishing at infinity) and then approximate $f$ by $r={ℒ}^{-1}R$. …

8: 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:

BibTeX

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9: 17.12 Bailey Pairs

… ►A sequence of pairs of rational functions of several variables $({\alpha }_n,{\beta }_n)$, $n=0,1,2,\mathrm{\dots }$, is called a Bailey pair provided that for each $n\geqq 0$ …

10: 31.8 Solutions via Quadratures

… ►By automorphisms from §31.2(v), similar solutions also exist for $m_0,m_1,m_2,m_3\in \mathbb{Z}$, and ${\mathrm{\Psi }}_{g,N}(\lambda ,z)$ may become a rational function in $z$. …