infinite products

SearchAdvancedHelp

(0.001 seconds)

1—10 of 50 matching pages

1: 4.36 Infinite Products and Partial Fractions

§4.36 Infinite Products and Partial Fractions

…

2: 4.22 Infinite Products and Partial Fractions

§4.22 Infinite Products and Partial Fractions

…

3: 5.8 Infinite Products

§5.8 Infinite Products

…

4: 23.17 Elementary Properties

… ►

§23.17(iii) Infinite Products

…

5: 1.10 Functions of a Complex Variable

… ►

§1.10(ix) Infinite Products

… ► ►

$M$-test

… ►

Weierstrass Product

… ►

§1.10(x) Infinite Partial Fractions

…

6: 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:

infiniteproduct, product representation

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

… ►

25.15.3 $$L(s,\chi )=k^{-s}\sum _{r=1}^{k-1}\chi (r)\zeta (s,\frac{r}{k}),$$

ⓘ

Symbols:

$L(s,\chi )$: Dirichlet $L$-function, $\zeta (s,a)$: Hurwitz zeta function, $k$: nonnegative integer, $s$: complex variable and $\chi (n)$: Dirichlet character

Keywords:

finite sum

Source:

Apostol (1976, (8), p. 255)

Notes:

In the original source the upper-index of the finite sum is $k$. We use $k-1$ since $\chi (k)=0$.

Referenced by:

(25.11.36), (25.11.36), §25.11(x), §25.15(i), Erratum (V1.0.23) for Equation (25.11.36)

Permalink:

http://dlmf.nist.gov/25.15.E3

Encodings:

pMML, png, TeX

See also:

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

…

7: 20.5 Infinite Products and Related Results

§20.5 Infinite Products and Related Results

… ►

8: 25.2 Definition and Expansions

… ►

§25.2(iv) Infinite Products

►

25.2.11 $$\zeta \left(s\right)=\prod _p{(1-p^{-s})}^{-1},$$ $\mathrm{\Re }s>1$,

ⓘ

Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\mathrm{\Re }$: real part, $p$: prime number and $s$: complex variable

Keywords:

infiniteproduct, primes

Source:

Apostol (1976, p. 231)

A&S Ref:

23.2.2

Referenced by:

§25.10(i)

Permalink:

http://dlmf.nist.gov/25.2.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(iv), §25.2 and Ch.25

… ►

25.2.12 $$\zeta \left(s\right)=\frac{{(2\pi )}^s{\mathrm{e}}^{-s-(\gamma s/2)}}{2(s-1)\mathrm{\Gamma }\left(\frac{1}{2}s+1\right)}\prod _{\rho }\left(1-\frac{s}{\rho }\right){\mathrm{e}}^{s/\rho },$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\gamma$: Euler’s constant, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $s$: complex variable and $\rho$: zeros

Keywords:

infiniteproduct, primes

Source:

Titchmarsh (1986b, (2.12.6), (2.12.8), p. 30–31)

A&S Ref:

23.2.10 (in slightly different form)

Permalink:

http://dlmf.nist.gov/25.2.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(iv), §25.2 and Ch.25

…

9: 5.17 Barnes’ $G$-Function (Double Gamma Function)

… ►

5.17.2 $$G\left(n\right)=(n-2)!(n-3)!\mathrm{\cdots }1!,$$ $n=2,3,\mathrm{\dots }$.

ⓘ

Symbols:

$G\left(z\right)$: Barnes’ $G$-function (or double gamma function), $!$: factorial (as in $n!$) and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/5.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.17 and Ch.5

…

10: 23.8 Trigonometric Series and Products

… ►

§23.8(iii) Infinite Products

…