Euler-product%20representation

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11: 32.8 Rational Solutions

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32.8.3 $$w(z;3)=\frac{3z^2}{z^3+4}-\frac{6z^2(z^3+10)}{z^6+20z^3-80},$$

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Symbols:

$z$: real

Permalink:

http://dlmf.nist.gov/32.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §32.8(ii), §32.8 and Ch.32

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$$Q_3(z)=z^6+20z^3-80,$$

… ►For determinantal representations see Kajiwara and Masuda (1999). … ►For determinantal representations see Kajiwara and Ohta (1998) and Noumi and Yamada (1999). … ►For determinantal representations see Masuda et al. (2002). …

12: Bibliography R

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A non-negative representation of the linearization coefficients of the product of Jacobi polynomials. Canad. J. Math. 33 (4), pp. 915–928.

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External Links:

ISSN 0008-414X, Document, Link, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.18(v).

Encodings:

BibTeX

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On the definition and properties of generalized $6$-$j$ symbols. J. Math. Phys. 20 (12), pp. 2398–2415.

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External Links:

ISSN 0022-2488, Document, MathReview (S. Saran), ZentralBlatt

Referenced by:

§16.4(iii), §16.4(iii).

Encodings:

BibTeX

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Integral representations for products of Airy functions. Z. Angew. Math. Phys. 46 (2), pp. 159–170.

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External Links:

ISSN 0044-2275, Document, MathReview (P. K. Banerji), ZentralBlatt

Referenced by:

§9.11(iii), §9.11(v), §9.11(v), (9.5.6), §9.5(ii).

Encodings:

BibTeX

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Integral representations for products of Airy functions. II. Cubic products. Z. Angew. Math. Phys. 48 (4), pp. 646–655.

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External Links:

ISSN 0044-2275, Document, MathReview (E. R. Love), ZentralBlatt

Referenced by:

(9.11.16), (9.11.17), §9.11(iii), §9.11(v).

Encodings:

BibTeX

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Integral representations for products of Airy functions. III. Quartic products. Z. Angew. Math. Phys. 48 (4), pp. 656–664.

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External Links:

ISSN 0044-2275, Document, MathReview (E. R. Love), ZentralBlatt

Referenced by:

§9.11(iii), §9.11(v).

Encodings:

BibTeX

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13: 8 Incomplete Gamma and Related
Functions

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14: 28 Mathieu Functions and Hill’s Equation

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15: 10.64 Integral Representations

§10.64 Integral Representations

… ►See Apelblat (1991) for these results, and also for similar representations for ${\mathrm{ber}}_{\nu }\left(x\sqrt{2}\right)$, ${\mathrm{bei}}_{\nu }\left(x\sqrt{2}\right)$, and their $\nu$-derivatives. …

16: 36.5 Stokes Sets

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36.5.4 $$80x^5-40x^4-55x^3+5x^2+20x-1=0,$$

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Symbols:

$x$: real parameter

Permalink:

http://dlmf.nist.gov/36.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

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36.5.7 $$X=\frac{9}{20}+20u^4-\frac{Y^2}{20u^2}+6u^2\mathrm{sign}\left(z\right),$$

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Symbols:

$\mathrm{sign}x$: sign of, $z$: real parameter, $X$: scaled coordinate and $Y$: scaled coordinate

Permalink:

http://dlmf.nist.gov/36.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

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17: 26.13 Permutations: Cycle Notation

… ►An explicit representation of $\sigma$ can be given by the $2\times n$ matrix: …

18: 8.26 Tables

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Khamis (1965) tabulates $P(a,x)$ for $a=0.05(.05)10(.1)20(.25)70$, $0.0001\le x\le 250$ to 10D.

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Abramowitz and Stegun (1964, pp. 245–248) tabulates $E_n\left(x\right)$ for $n=2,3,4,10,20$, $x=0(.01)2$ to 7D; also $(x+n){\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2,3,4,10,20$, $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

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Pagurova (1961) tabulates $E_n\left(x\right)$ for $n=0(1)20$, $x=0(.01)2(.1)10$ to 4-9S; ${\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2(1)10$, $x=10(.1)20$ to 7D; ${\mathrm{e}}^x{E}_p\left(x\right)$ for $p=0(.1)1$, $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

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Zhang and Jin (1996, Table 19.1) tabulates $E_n\left(x\right)$ for $n=1,2,3,5,10,15,20$, $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

19: 23 Weierstrass Elliptic and Modular
Functions

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20: Bibliography G

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Integral representations for computing real parabolic cylinder functions. Numer. Math. 98 (1), pp. 105–134.

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External Links:

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

Referenced by:

§12.18.

Encodings:

BibTeX

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Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

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Notes:

Includes Fortran program claiming 12S accuracy

External Links:

ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt

Referenced by:

3rd item, §10.77(ix).

Encodings:

BibTeX

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Stirling number representation problems. Proc. Amer. Math. Soc. 11 (3), pp. 447–451.

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External Links:

FullText, MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§26.1, §26.1, Notations S, Notations S.

Encodings:

BibTeX

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Bessel functions and representation theory. I. J. Functional Analysis 22 (2), pp. 73–105.

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External Links:

Document, MathReview (E. A. Thieleker), ZentralBlatt

Referenced by:

§35.5(i).

Encodings:

BibTeX

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Representations of Integers as Sums of Squares. Springer-Verlag, New York.

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External Links:

ISBN 0-387-96126-7, MathReview (Harvey Cohn), ZentralBlatt

Referenced by:

§27.13(iv).

Encodings:

BibTeX

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