product representation

SearchAdvancedHelp

(0.001 seconds)

1—10 of 57 matching pages

1: 27.4 Euler Products and Dirichlet Series

… ►The completely multiplicative function $f(n)=n^{-s}$ gives the Euler product representation of the Riemann zeta function $\zeta \left(s\right)$ (§25.2(i)): …

2: 25.15 Dirichlet $L$-functions

… ►

25.15.2 $$L(s,\chi )=\prod _p{\left(1-\frac{\chi (p)}{p^s}\right)}^{-1},$$ $\mathrm{\Re }s>1$,

ⓘ

Symbols:

$L(s,\chi )$: Dirichlet $L$-function, $\mathrm{\Re }$: real part, $p$: prime number, $s$: complex variable and $\chi (n)$: Dirichlet character

Keywords:

infinite product, productrepresentation

Source:

Apostol (1976, p. 231)

Permalink:

http://dlmf.nist.gov/25.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.15(i), §25.15 and Ch.25

… ►

25.15.4 $$L(s,\chi )=L(s,{\chi }_0)\prod _{p|k}\left(1-\frac{{\chi }_0(p)}{p^s}\right),$$

ⓘ

Symbols:

$L(s,\chi )$: Dirichlet $L$-function, $k$: nonnegative integer, $p$: prime number, $s$: complex variable and $\chi (n)$: Dirichlet character

Keywords:

productrepresentation

Source:

Apostol (1976, p. 262)

Referenced by:

§25.15(i)

Permalink:

http://dlmf.nist.gov/25.15.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §25.15(i), §25.15 and Ch.25

…

3: 25.10 Zeros

… ►The product representation (25.2.11) implies $\zeta \left(s\right)\ne 0$ for $\mathrm{\Re }s>1$. …

4: Bibliography R

… ►

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

ⓘ

External Links:

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

Referenced by:

§18.18(v).

Encodings:

BibTeX

… ►

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

ⓘ

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

►

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

ⓘ

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

►

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

ⓘ

External Links:

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

Referenced by:

§9.11(iii), §9.11(v).

Encodings:

BibTeX

…

5: 10.32 Integral Representations

… ►

§10.32(iii) Products

… ►For collections of integral representations of modified Bessel functions, or products of modified Bessel functions, see Erdélyi et al. (1953b, §§7.3, 7.12, and 7.14.2), Erdélyi et al. (1954a, pp. 48–60, 105–115, 276–285, and 357–359), Gröbner and Hofreiter (1950, pp. 193–194), Magnus et al. (1966, §3.7), Marichev (1983, pp. 191–216), and Watson (1944, Chapters 6, 12, and 13).

6: 10.9 Integral Representations

… ►

§10.9(iii) Products

… ► …

7: 9.11 Products

… ►

§9.11(iii) Integral Representations

… ►For an integral representation of the Dirac delta involving a product of two $\mathrm{Ai}$ functions see §1.17(ii). …

8: 18.17 Integrals

… ►

§18.17(ii) Integral Representations for Products

►

Ultraspherical

… ►

Legendre

… ►For addition formulas corresponding to (18.17.5) and (18.17.6) see (18.18.8) and (18.18.9), respectively. …

9: Bibliography V

… ►

Integral representations for products of Lamé functions by use of fundamental solutions. SIAM J. Math. Anal. 15 (3), pp. 559–569.

ⓘ

External Links:

ISSN 0036-1410, Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§29.8.

Encodings:

BibTeX

…

10: 13.12 Products

… ►For integral representations, integrals, and series containing products of $M(a,b,z)$ and $U(a,b,z)$ see Erdélyi et al. (1953a, §6.15.3).